Search Results for Ring

Biographies

  1. Herstein biography
    • The first of these papers generalised the theorem of Wedderburn that shows that every finite division ring is commutative.
    • The second paper proves a conjecture that the solubility of groups of odd order is equivalent to a condition on the group ring of a group, while the third paper takes methods from the study of Banach rings and topological groups to prove results about group rings over the complex numbers.
    • During this time he worked on a topic which was to be one of the main themes of his work, namely on conditions on a ring which imply commutativity.
    • If R is a ring with centre C, and if xn - x is in C for all x in R, n a fixed integer larger than 1, then R is commutative.
    • In particular he examined finite subgroups of a division ring.
    • Other algebra books included a more advanced ring theory book Noncommutative rings (1968) and Topics in ring theory (1969).
    • This colourful and informative book on noncommutative ring theory is based on a series of expository lectures given by the author in the summer of 1965 at Bowdoin College before an audience of teachers from colleges and small universities.
    • These lectures were in turn based to a large extent on the author's 1961 and 1965 University of Chicago notes on ring theory.
    • The spirit of the Carus Monograph series is clearly embodied in this moving and excellently written account of important aspects of classical and modern ring theory.
    • Topics in Ring Theory was based on lectures Herstein gave at the University of Chicago and first published in the University of Chicago Mathematics Lecture Notes series.

  2. Krull biography
    • Ring theory results from this thesis have recently been found important in the area of coding theory.
    • In 1928 he defined the Krull dimension of a commutative Noetherian ring and brought ring theory into in new setting in which he was able to show that the principal ideal theorem held.
    • was quickly recognised as a decisive advance in Noether's programme of emancipating abstract ring theory from the theory of polynomial rings.
    • Krull carried his work forward, defining further concepts which are today central to modern research in ring theory.
    • He then wrote the remarkable treatise Ideal Theory which remains a beautiful introduction to ring theory but is simply a theory built from the results that Krull had himself proved.
    • Another major topic in ring theory is the study of local rings, that is rings having a unique maximal ideal, and they are used in the study of local properties of algebraic varieties.
    • Indeed much of modern ring theory is still following the path which Krull took, building on the foundations which Emmy Noether had laid.
    • History Topics: The development of Ring Theory .

  3. Lasker biography
    • He proved the primary decomposition theorem for an ideal of a polynomial ring in terms of primary ideals in a paper Zur Theorie der Moduln und Ideale published in volume 60 of Mathematische Annalen in 1905.
    • A commutative ring R is now called a 'Lasker ring' if every ideal of R can be represented as an intersection of a finite number of primary ideals.
    • Nathan Divinsky, himself an exceptional mathematician and like Lasker most famous for his results in ring theory, writes:- .
    • Finally let us comment that Lasker's results on the decomposition of ideals into primary ideals was the foundation on which Emmy Noether built an abstract theory which developed ring theory into a major mathematical topic and provided the foundations of modern algebraic geometry.
    • Emmy Noether's Idealtheorie in Ringbereichen (1921) was of fundamental importance in the development of modern algebra, generalising Lasker's results by giving the decomposition of ideals into intersections of primary ideals in any commutative ring with ascending chain condition.
    • History Topics: The development of Ring Theory .

  4. Pless biography
    • Both Emmy Noether and Irving Kaplansky were algebraists, particularly studying ring theory, so it is not surprising that their inspiration led her to algebra.
    • A publication arising from her doctoral work was The continuous transformation ring of biorthogonal bases spaces which was published in the Duke Mathematical Journal in the following year.
    • In this paper Pless looks at dual vector spaces of countable dimension over a division ring and studies the ring L of all continuous linear transformations on such a space.
    • The main results describe the structure of B/A where B and A are non-zero ideals in L proving, in particular, that B/A is a primitive ring that is not regular.

  5. Gemma Frisius biography
    • In 1534 Gemma Frisius published Tractatus de Annulo Astronomicae in which he described an instrument he called the astronomer's ring which he had designed and Van der Heyden had made in his workshop.
    • Gemma states that the astronomer's ring was (see for example [Bibliotheca universalis, Zurich, 1545 (Karrow, 1993).',3)">3]):- .
    • [I have] augmented the ring so much that from simply showing the hours of the day and the four directions it now rivals whatever mathematical instruments you will.
    • [Many ideas of others are] brought together into this single ring.
    • the first astronomer's staff in brass that was devised by Gemma Frisius; the two great globes of Gerardus Mercator; and the astronomer's ring of brass as Gemma Frisius had newly framed it.

  6. Redei biography
    • Between 1936 and 1942 he looked at the problem of determining which real quadratic number fields Q(√d) have a ring of integers which is a Euclidean ring.
    • In these he found several of the 21 cases, also showing that many others do not have a Euclidean ring of integers.
    • He did, however, get one of these wrong for he 'proved' in the last of the three papers we mentioned that Q(√97) has a ring of integers which is a Euclidean ring.

  7. Goldie Alfred biography
    • Hirsch advised Goldie to concentrate on an area of algebra other than universal algebra, so he decided that ring theory would provide a good area which would complement the active UK research area of group theory which was led by Philip Hall.
    • Deciding that he needed to carve out an area of his own if he was to make a success of the academic profession, Goldie started to undertake research on his own in ring theory [Amer.
    • One of the most characteristic features of ring theory in the late 1940s and early 1950s was the search for a structure theory for general rings.
    • Thus it is not really surprising that Goldie's early work in ring theory was influenced by Jacobson's papers and books.
    • At Leeds he continued his work in algebra, building a "school" in ring theory, hosting many visitors, and organizing memorable conferences.

  8. Cohn biography
    • In research interests Cohn has worked widely in many areas of algebra but, in particular he has made outstanding contributions to non-commutative ring theory.
    • Over the next few years his work ranged across group theory, field theory, Lie rings, semigroups, abelian groups and ring theory.
    • From the mid 1960s his work concentrates on non-commutative ring theory and the theory of algebras.
    • It completes the formation of the theory of free associative algebras and related classes of rings as an independent domain of ring theory.

  9. Shoda biography
    • Here the multiplication is of course associative, and distributive for the latter compositions, presenting the notion of ring-systems as a generalization of the ring notion.
    • Structural theory of abstract ring-systems is developed, under chain conditions, including (generalized) Peirce decompositions and Wedderburn's theorem; for the latter the notion of matrices is also generalized.
    • Particular observations are made to the cases of ring-systems, rings, and groups.

  10. Huygens biography
    • By 1656 Huygens was able to confirm his ring theory to Boulliau and the results were reported to the Paris group.
    • In Systema Saturnium (1659), Huygens explained the phases and changes in the shape of the ring.
    • However by 1665 even Fabri was persuaded to accept Huygens' ring theory as improving telescopes confirmed his observations.
    • Christiaan Huygens' article on Saturn's Ring .

  11. Dedekind biography
    • It was in the third and fourth editions of Vorlesungen uber Zahlentheorie, published in 1879 and 1894, that Dedekind wrote supplements in which he introduced the notion of an ideal which is fundamental to ring theory.
    • Dedekind formulated his theory in the ring of integers of an algebraic number field.
    • The general term 'ring' does not appear, it was introduced later by Hilbert.
    • History Topics: The development of Ring Theory .

  12. Amitsur biography
    • the editors divide Amitsur's work into four main areas: general ring theory, structure theory of rings with polynomial identities, combinatorial theory of PI-algebras and theory of division algebras.
    • Vesselin Drensky, reviewing the work, writes about the results on general ring theory:- .
    • Embarking on his mathematical career at the time when ring theorists were searching for a general structure theory in the spirit of invariant theory, representation theory and theory of finite-dimensional algebras, Amitsur breathed life into the new theory and developed a body of theorems which were to provide inspiration for a generation of ring theorists.

  13. Mochizuki biography
    • he went to University of Washington where he studied ring theory for his doctorate under the supervision of James P Jans.
    • Mochizuki's first three papers were on ring theory.
    • For anyone interested let us record that a QF-3 ring is defined as a ring R which, considered as a left R-module, can be imbedded in a projective injective left R-module Q.

  14. Helmholtz biography
    • If they both have the same direction of rotation they will proceed in the same sense, and the ring in front will enlarge itself and move slower, while the second one will shrink and move faster, if the velocities of translation are not too different, the second will finally reach the first and pass through it.
    • Then the same game will be repeated with the other ring, so the ring will pass alternately one through the other.

  15. Foster biography
    • Dieudonne explains what Foster means by a Boolean-like ring:- .
    • A "Boolean-like" ring is a commutative ring H with unit element such that a + a = 0 for all a in H and ab(a + b + ab) = ab for any two elements a, b of H.

  16. Noether Emmy biography
    • At Gottingen, after 1919, Noether moved away from invariant theory to work on ideal theory, producing an abstract theory which helped develop ring theory into a major mathematical topic.
    • In this paper she gave the decomposition of ideals into intersections of primary ideals in any commutative ring with ascending chain condition.
    • History Topics: The development of Ring Theory .

  17. Chow biography
    • Such equivalence classes make up the "Chow ring" of a nonsingular projective variety and provide the algebraic counterpart of the topological singular cohomology ring.
    • The "Chow ring" is just as fundamental in algebraic geometry as its topological counterpart.

  18. Fermat biography
    • The following report, made to Colbert the leading figure in France at the time, has a ring of truth:- .
    • Unsuccessful attempts to prove the theorem over a 300 year period led to the discovery of commutative ring theory and a wealth of other mathematical discoveries.
    • History Topics: The development of Ring Theory .

  19. Northcott biography
    • No previous knowledge whatsoever of ring theory is assumed, and beginners to the subject will find here a very readable account.
    • He has assumed that the reader is familiar with the notions of group, ring, and field, but otherwise the presentation is self-contained.
    • It focuses on the construction of the tensor, exterior and symmetric algebras of a module over a commutative ring and, by bringing out some of their relationships, develops the theory of several associated structures.

  20. Jacobson biography
    • Jacobson is well known for his outstanding contributions to ring theory.
    • Jacobson discovered a deep structure theory for rings and has given his name to the Jacobson radical, the intersection of the maximal ideals of a ring.
    • History Topics: The development of Ring Theory .

  21. Tait biography
    • He claimed that two interacting rings would change size and velocity as they interacted but would retain their ring shape.
    • We sometimes can make one ring shoot through another, illustrating perfectly your description; when one ring passes near another, each is much disturbed, and is seen to be in a state of violent vibration for a few seconds, till it settles again into its circular form.

  22. Malcev biography
    • In 1937 Malcev published a paper on the embeddability of a ring in a field, answering a question posed by Kolmogorov.
    • Malcev answered this question by constructing a ring whose multiplicative semigroup was not embeddable in a group.
    • He was led to investigate the existence of rings whose multiplicative semigroup was embeddable in a group yet the ring still was not embeddable in a field.

  23. Bliss biography
    • Bliss received his doctorate in 1900 for a dissertation The Geodesic Lines on the Anchor Ring which was supervised by Bolza.
    • Bliss published two papers in 1902: one in the Annals of Mathematics was based on his doctoral dissertation and had the same title The geodesic lines on the anchor ring while the second in the Transactions of the American Mathematical Society was titled The second variation of a definite integral when one end-point is variable.

  24. Euler biography
    • Although there were problems with his approach this eventually led to Kummer's major work on Fermats Last Theorem and to the introduction of the concept of a ring.
    • History Topics: The development of Ring Theory .

  25. Hammer biography
    • Let R denote an infinite associative ring with unity, with the property that every proper factor ring of R is finite.

  26. Alberti biography
    • Polyalphabetic substitution was introduced into diplomatic practice by Alberti, who also invented a simple mechanical device to speed up coding and decoding, consisting of a fixed and a movable ring.
    • excelled in all bodily exercises; could, with feet tied, leap over a standing man; could in the great cathedral, throw a coin far up to ring against the vault; amused himself by taming wild horses and climbing mountains.

  27. Ore biography
    • He then worked on non-commutative ring theory and proved his celebrated embedding theorem for a non-commutative integral domain into a division ring.

  28. Hoehnke biography
    • Hoehnke's early papers include Uber die definierenden Gleichungen fur Matrizeneinheiten in primaren Ringen (1956) in which he looks at multiplicative semigroups inside matrix rings over completely primary rings, Identische Kongruenzen fur Polynome nach zusammengesetzten Moduln (1956), and Nilpotenzkriterien (1957) in which he looks at conditions on a ring which force certain radicals to be nilpotent.
    • The bulk of Hoehnke's work on semigroups relies on ring theory and is based on the observation that, in most aspects, congruences of semigroups play the role of ideals of rings.

  29. Seidenberg biography
    • When does the Lasker-Noether decomposition theorem, which says that an ideal in a commutative Noetherian ring is the intersection of a finite number of primary ideals, hold in a constructive sense? .
    • In the paper he gives conditions on the ring R so that given generators for an ideal in a R[x1, ..

  30. Kummer biography
    • Not only has his work been most fundamental in work relating to Fermat's Last Theorem, since all later work was based on it for many years, but the concept of an ideal allowed ring theory, and much of abstract algebra, to develop.
    • History Topics: The development of Ring Theory .

  31. Bennett biography
    • In the first part of this 150 page paper he examines, in modern terminology, the group of units of the ring of integers modulo m.
    • A system for an aircraft compass comprises a skeleton hemispherical dome constructed of ribs carrying the pivot at the summit and connected by a ring with carriers each holding three magnets.

  32. Maxwell biography
    • I have effected several breaches in the solid ring, and now am splash into the fluid one, amid a clash of symbols truly astounding.
    • When I reappear it will be in the dusky ring, which is something like the siege of Sebastopol conducted from a forest of guns 100 miles one way, and 30,000 miles the other, and the shot never to stop, but go spinning away round a circle, radius 170,000 miles..

  33. Thompson Robert biography
    • [Thompson's] purpose in this paper is to develop the theory of unimodular equivalence for matrices whose entries come from the Hurwitz ring of integral quaternions.
    • This is done in complete detail, and a normal form is obtained which exhibits as much uniqueness as is possible for the case of a noncommutative ring.

  34. Cassini biography
    • He discovered the gap in the ring system of Saturn now known as the Cassini division in 1675.
    • Cassini and the Division in Saturn's Ring .

  35. Wedderburn biography
    • In this paper On hypercomplex numbers which appeared in the Proceedings of the London Mathematical Society, he showed that every semisimple algebra is a direct sum of simple algebras and that a simple algebra was a matrix algebra over a division ring.
    • History Topics: The development of Ring Theory .

  36. Quillen biography
    • In the 1960s, Quillen described how to define the homology of simplical objects over many different categories, including sets, algebras over a ring, and unstable algebras over the Steenrod algebra.
    • He received the award as the principal architect of the higher algebraic K-theory in 1972, a new tool that successfully used geometric and topological methods and ideas to formulate and solve major problems in algebra, particularly ring theory and module theory.

  37. Servois biography
    • Although he does not use the concept of a ring explicitly, he does verify that linear commutative operators satisfy the ring axioms.

  38. Zorn biography
    • He studied the structure of semisimple alternative rings in 1932, proving that such a ring is a direct sum of simple alternative algebras which he classified.
    • In Alternative rings and related questions I: existence of the radical published in 1941 Zorn considered the theory of the radical of an alternative ring.

  39. Kolchin biography
    • But in fact the foundational aspects of the theory are better served by an abstract view of algebraic varieties as sets with an additional structure, where the latter is usually a sheaf of integral domains, but could be taken as a collection of fields with specialization relations (the ring of sections over a set is replaced by its quotient field and the restriction maps endue the specializations).

  40. Moser William biography
    • Moser published a fine collection of combinatorics papers jointly with Morton Abramson: A note on combinations (1966); Combinations, successions and the n-kings problem (1966); Permutations without rising or falling w-sequences (1967); Enumeration of combinations with restricted differences and cospan (1969); Generalizations of Terquem's problem (1969); The problem of the second seating and generalizations (1972); Arrays with fixed row and column sums (1973); and Linear and ring arrangements (1976).

  41. Grauert biography
    • This text is an excellent introduction to the classical themes of modern several complex variables theory: domains of holomorphy, holomorphic complexity, pseudoconvexity, the ring of convergent power series, analytic subvarieties and the several variables version of the Mittag-Leffler and Weierstrass problems ..

  42. Fiorentini biography
    • In these the authors explain the meaning of the basic notions (e.g., truth-table valuation, formal proof, set, relation, cardinal number, well ordering, axiom of choice, associative law, group, ring, field, isomorphism) and state some of their properties, but give no proofs except in the very simplest cases.

  43. Liouville biography
    • History Topics: The development of Ring Theory .

  44. Moufang biography
    • She proves that the rational group algebra of this group can be embedded in an ordered division ring.

  45. Samuel biography
    • These devices have been developed by various authors: W Krull, who introduced the fundamental notion of a local ring, O Zariski, I Cohen, P Samuel, and C Chevalley.

  46. Munn biography
    • His discovery of Passman's books on infinite group rings brought about a further change in the main thrust of his work, and in the eighties, while still writing the occasional paper on pure semigroup theory, he returned to the study of semigroup algebras, publishing a series of remarkable papers linking semigroup properties to ring-theoretic properties of their algebras.

  47. Kostrikin biography
    • the calculation of the number of unitary conservative complex polynomials and the determination of an upper bound in terms of p for the nilpotence class of a Lie ring having a fixed-point free automorphism of prime order.

  48. Lissajous biography
    • The demonstration began with a 'sheaf of light' thrown from the lamp on to a mirror held in the lecturer's hand; when he moved the mirror quickly he could produce a ring of light on the ceiling and various other figures, thus illustrating the persistence of vision that is an essential feature of many of the remainder of the demonstrations.

  49. Airey biography
    • John R Airey, The Lommel-Weber ˝ Function and Its Application to the Problem of Electric Waves on a Thin Anchor Ring, Proc.

  50. Galileo biography
    • Continued observations were puzzling indeed to Galileo as the bodies on either side of Saturn vanished when the ring system was edge on.

  51. Higman biography
    • Higman also worked on topics such as: varieties of groups; enumerating p-groups; and Lie ring methods for finite nilpotent groups.

  52. Delamain biography
    • If we think of modern instruments as computers then one would have to say that Delamain's views have a ring of realism in today's world which are somewhat lacking in Oughtred's high ideals.

  53. Hurwitz biography
    • This involves studying the ring of integer quaternions in which there are 24 units.

  54. Gordan biography
    • History Topics: The development of Ring Theory .

  55. Gruenberg biography
    • On the other hand, I hoped to persuade ring theorists that here was an area of group theory well suited to applications of integral representation theory.

  56. Kolmogorov biography
    • Another contribution of the highest significance in this area was his definition of the cohomology ring which he announced at the International Topology Conference in Moscow in 1935.

  57. Ribenboim biography
    • Instead of the customary material on ideal factorization and unit theorems, the reader will find such topics as p-adic logarithms, the Witt ring, infinite Galois theory, ordering of number fields and diophantine dimension (including Terjanian's celebrated counterexample).

  58. Jung biography
    • The analytic approach, followed in the present book, uses as fundamental concept that of a place of the field K, a place being an isomorphism of K into the quotient field of the ring of convergent power series in two variables (the uniformizing variables at the place) with the requirement that distinct couples of values of the variables u, v in these power series sufficiently near to (0, 0) should lead to distinct values for some function of the field.

  59. Kaplansky biography
    • He has made major contributions to ring theory, group theory and field theory.

  60. Lefschetz biography
    • He made extensive use of product spaces; he developed intersection theory, including the theory of the intersection ring of a manifold; and he made essential contributions to various kinds of homology theory, notably relative homology, singular homology, and cohomology.

  61. Hippias biography
    • to have gone once to the Olympian festival with everything that he wore made by himself, ring and sandal (engraved), oil-bottle, scraper, shoes, clothes, and a Persian girdle of expensive type; he also took poems, epics, tragedies, dithyrambs, and all sorts of prose works.

  62. Szekeres biography
    • He played in the North Sydney Symphony Orchestra and the Ku-ring-gai Philharmonic Orchestra, also supporting this orchestra by acting as treasurer from the time it was founded until 2000.

  63. Kloosterman biography
    • The group he studied was the special linear group of 2 by 2 matrices over the ring of integers modulo pn.

  64. Chen biography
    • He published Integration in free groups (1951), Commutator calculus and link invariants (1952), Isotopy invariants of links (1952), and A group ring method for finitely generated groups (1954).

  65. Kronecker biography
    • History Topics: The development of Ring Theory .

  66. Deuring biography
    • I have succeeded in doing so, all the way to the construction of the ring of multipliers and the proof that it is algebraic.

  67. Magnus biography
    • During this period Magnus introduced Lie ring methods to study the lower central series of free groups.

  68. Oughtred biography
    • His Grammelogia, or the Mathematicall ring was published in 1630.

  69. Tutte biography
    • In 1946 he published On Hamiltonian circuits, and in the following year the two papers A family of cubical graphs and A ring in graph theory.

  70. Magnitsky biography
    • He went from there to the Simonov Monastery, a fortified monastery built as part of an outer fortification ring for the city of Moscow, where he trained to be a Russian Orthodox priest.

  71. Zaanen biography
    • Links with pure lattice theory and ring theory are also explored.

  72. Steinitz biography
    • In his talk Steinitz introduced an algebra over the ring of integers whose base elements are isomorphism classes of finite abelian groups.

  73. Van der Waerden biography
    • History Topics: The development of Ring Theory .

  74. Bass biography
    • The Grothendieck group of projective modules over a ring leads to K0 .

  75. Adams Edwin biography
    • In 1920 he published The potential of ring-shaped discs.

  76. Mazur Barry biography
    • Wiles's proof of the Taniyama-Shimura conjecture and of Fermat's Last Theorem, in "Modular elliptic curves and Fermat's last theorem" (1995), using results with R Taylor in "Ring-theoretic properties of certain Hecke algebras".

  77. Copson biography
    • However he always turned it off when he lectured and alarm clocks were sometimes brought into the lecture room and would ring loudly at various times.

  78. Schwarz Stefan biography
    • Continuing his research, he studied arithmetic in the ring of integers in algebraic number fields.

  79. Pic biography
    • The final chapter is on rings and discusses: definition of a ring; ideals; rings of fractions; characteristics of rings; products of rings; polynomial rings; symmetric polynomials and symmetric rational functions; divisibility in integral domains; prime ideals and prime radicals of ideals in associative rings; Artinian and Noetherian rings; Dedekind domains; primary ideals in associative and commutative rings with unity; algebraic varieties; and the Jacobson radical in associative rings with identity.

  80. Wexler-Kreindler biography
    • The object of this note is to include in the language of the theory of abelian categories some results known in the theory of modules over principal rings, in particular the theorem concerning the submodules of free modules over principal rings, as well as the theorem on the invariants factors of a submodule of finite type of a free module over a principal ring (non-commutative).

  81. Bachet biography
    • History Topics: The development of Ring Theory .

  82. Chernikov biography
    • He also studied finiteness type conditions that had already been seen to have great importance in ring theory, namely finiteness type conditions which did not allow infinite chains of subgroups of a specified type.

  83. Blum biography
    • An important first contribution was Blum's 1989 paper Lectures on a theory of computation and complexity over the reals (or an arbitrary ring) which extended the theories of computation and computational complexity from the standard discrete situation to study how these ideas can be developed in continuous domains such as the real number system.

  84. Hazlett biography
    • She enjoyed telling us about one of her papers whose title she gave as "Embedding a ring in a field," and she enjoyed telling us how her colleague Shaw teased her about it - it evoked a picture of nefarious agricultural activities, he said.

  85. Diophantus biography
    • History Topics: The development of Ring Theory .

  86. Arf biography
    • In ring theory, Arf rings are named after him.

  87. Kurosh biography
    • Gradually, along with papers on group theory, Kurosh began to publish papers on ring theory, linear algebra and lattices; later, also papers on category theory and the theory of multi-operator groups, rings and linear algebras.

  88. Cayley biography
    • History Topics: The development of Ring Theory .

  89. Loewy biography
    • Such ideas clearly influenced Fraenkel to introduce the notion of a ring, and in particular zero-divisors in rings.

  90. Weber biography
    • History Topics: The development of Ring Theory .

  91. Remak biography
    • He was particularly interested in the exciting new mathematical developments which were written up in van der Waerden's two volume Algebra published in 1930 which contained the new developments in ring theory by Emmy Noether, Hilbert, Dedekind and Artin.

  92. Clavius biography
    • Taking Clavius' account of the eclipse of 1567 at face value, it might perhaps seem that he witnessed the ring phase of an annular eclipse.

  93. Mackey biography
    • Early in his career Mackey worked on the duality theory of locally convex spaces publishing papers which include On infinite dimensional linear spaces (1943), On convex topological linear spaces (1943), Equivalence of a problem in measure theory to a problem in the theory of vector lattices (1944), (with Shizuo Kakutani) Ring and lattice characterization of complex Hilbert space (1946), On convex topological linear spaces (1946).

  94. Steinfeld biography
    • Basically he was seeking appropriate analogues in ring theory for certain concepts used in the theory of groups and he did this by looking at the corresponding notions in semigroups.

  95. Macaulay biography
    • In 1915 Macaulay discovered the primary decomposition of an ideal in a polynomial ring, which is the analogue of the decomposition of a number into a product of prime powers.

  96. Al-Khazin biography
    • using a ring of about 4 meters.

  97. Hamilton biography
    • History Topics: The development of Ring Theory .

  98. Frattini biography
    • History Topics: The development of Ring Theory .

  99. Hopf biography
    • His work on the homology of manifolds, undertaken in Princeton in 1927-28, led to his definition of the intersection ring by defining a product on cycles by their intersection.

  100. Gelfand biography
    • They showed that these rings could always be represented as a ring of linear operators on a Hilbert space.

  101. Legendre biography
    • History Topics: The development of Ring Theory .

  102. Barnes biography
    • Undaunted by this set-back, Barnes returned to his accusations on the cement ring in a speech he delivered in the House of Lords the following year, in which he claimed that powerful business concerns were using libel and slander action to suppress criticism.

  103. Rado Ferenc biography
    • He continued with non-injective collineations of two Desarguesian projective planes, which led him to ring geometry.

  104. Lame biography
    • History Topics: The development of Ring Theory .

  105. Artin biography
    • History Topics: The development of Ring Theory .

  106. Barrow biography
    • Barrow was considered to be the ring leader of a group of royalists from 1648.

  107. Clarke Joan biography
    • Clarke was formally introduced to Alan Turing's family and vice versa, he gave her an engagement ring, although she did not wear it when in the Hut, choosing to keep their engagement secret from their colleagues.

  108. Hilbert biography
    • History Topics: The development of Ring Theory .

  109. Molin biography
    • Molien introduced the idea of a group ring in his study of group representations.

  110. Autolycus biography
    • Again eclipses of the sun were sometimes total, sometimes annular where moon appears smaller than the sun and a ring of the sun is visible right round the moon.

  111. Stampioen biography
    • Was Stampioen correct? Well if we look at what he was trying to do, it was to do his arithmetic in the ring Z[√b].

  112. Cauchy biography
    • History Topics: The development of Ring Theory .

  113. Arago biography
    • Craters on both the Moon and Mars have named after him, as has a ring of Neptune.

  114. Gauss biography
    • History Topics: The development of Ring Theory .

  115. Dirichlet biography
    • History Topics: The development of Ring Theory .

  116. Knuth biography
    • A semifield is an algebraic structure satisfying all the usual axioms for a division ring except associativity of multiplication.

History Topics

  1. Ring Theory
    • The development of Ring Theory .
    • Any book on Abstract Algebra will contain the definition of a ring.
    • It will define a ring to be a set with two operations, called addition and multiplication, satisfying a collection of axioms.
    • A ring is therefore a setting for generalising integer arithmetic.
    • What motivated this abstract definition of a ring? .
    • Our comment above that study of a ring provided a generalisation of integer arithmetic is the clue to the early development of commutative ring theory.
    • Euler's work on the case n = 3 involved extending ordinary integer arithmetic to apply to the ring of numbers of the form a + b√-3 where a, b are integers.
    • However, Euler failed to grasp the difficulties of working in this ring and made certain assertions which, although true, would be hard to justify.
    • Complex numbers of the form a + b√-3, where a, b are integers, form a ring.
    • A prime number in this ring is defined in an analogous way to a prime integer, namely a number whose only divisors of the form a + b√-3 other than itself are those numbers with multiplicative inverses.
    • In this ring 4 can be written as a product of prime numbers in two different ways .
    • Dedekind defined an "ideal", characterising it by its now familiar properties: namely that of being a subring whose elements, on being multiplied by any ring element, remain in the subring.
    • Ring theory in its own right was born together with an early hint of the axiomatic method which was to dominate algebra in the 20thCentury.
    • Dedekind also introduced the word "module" (early spelling: "modul") in 1871 although its initial definition was considerably more restricted than the present definition, being first introduced as a subgroup of the additive group of a ring; the term is now used for a "vector space with coefficients from a ring".
    • For example all integers divisible by a fixed prime p form a prime ideal of the ring of integers.
    • Although the concept of a ring is due to Dedekind, one of the first words used was an "order" or "order-modul".
    • Dedekind did introduce the term "field" (Korper) for a commutative ring in which every non-zero element has a multiplicative inverse but the word "number ring" (Zahlring) or "ring" is due to Hilbert.
    • In contrast to commutative ring theory, which as we have seen grew from number theory, non-commutative ring theory developed from an idea which, at the time of its discovery, was heralded as a great advance in applied mathematics.
    • The quaternion algebra, as Hamilton called this four dimensional algebra, was widely used in applied mathematics (where it was later replaced by the vector product) and it launched non-commutative ring theory.
    • Matrices with their laws of addition and multiplication were introduced by Cayley in 1850 while, in 1870, Pierce noted that the now familiar ring axioms held for square matrices - another early example of the axiomatic approach to rings.
    • In 1905 he proved that every finite division ring (a ring in which every non-zero element has a multiplicative inverse) is commutative and so is a field.
    • In 1908 Wedderburn had the important idea of splitting the study of a ring into two parts, one part he called the radical, the part which was left being called semi-simple.
    • It is interesting to note that this fundamental work by Jacobson hinges on the idea of the "Jacobson radical" of a ring which is an analogue of a group theory idea due to Frattini as early as 1885.
    • http://www-history.mcs.st-andrews.ac.uk/HistTopics/Ring_theory.html .

  2. Ring Theory references
    • References for: The development of Ring Theory .
    • http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Ring_theory.html .

  3. Ring Theory references
    • References for: The development of Ring Theory .
    • [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Ring_theory.html] .

  4. Knots and physics
    • If they both have the same direction of rotation they will proceed in the same sense, and the ring in front will enlarge itself and move slower, while the second one will shrink and move faster, if the velocities of translation are not too different, the second will finally reach the first and pass through it.
    • Then the same game will be repeated with the other ring, so the ring will pass alternately one through the other.
    • We sometimes can make one ring shoot through another, illustrating perfectly your description; when one ring passes near another, each is much disturbed, and is seen to be in a state of violent vibration for a few seconds, till it settles again into its circular form.

  5. Mathematical games
    • The rings were arranged so that only the ring A at one end could be taken on and off without problems.
      Go directly to this paragraph
    • To take any other off the ring next to it towards A had to be on the bar and all others towards A had to be off the bar.
      Go directly to this paragraph
    • In fact Lucas (the inventor of the Towers of Hanoi) gives a pretty solution to Cardan's Ring Puzzle using binary arithmetic.
      Go directly to this paragraph

  6. Nine chapters references
    • S S Bai, A re-examination of a ring area problem in the 'Jiu zhang suanshu' (Chinese), Beijing Shifan Daxue Xuebao 30 (1) (1994), 139-142.
    • J M Li, A textual criticism on the "art of milu" in ring measurement in Nine chapters on arithmetic (Chinese), J.

  7. Bourbaki 2
    • Everything is done for abstractly-valued functions whose values lie in a topologized vector-space, or, at the very least, in a normed ring, but there are many fine features of analysis spread out underneath this superimposed layer of ever-present generalizations.
    • The second of the two chapters covers the theory of modules over a principal ideal ring.

  8. Nine chapters references
    • S S Bai, A re-examination of a ring area problem in the 'Jiu zhang suanshu' (Chinese), Beijing Shifan Daxue Xuebao 30 (1) (1994), 139-142.
    • J M Li, A textual criticism on the "art of milu" in ring measurement in Nine chapters on arithmetic (Chinese), J.

  9. Orbits
    • Le Verrier was convinced that a planet or ring of material lay inside the orbit of Mercury but being close to the Sun had not been observed.
      Go directly to this paragraph

  10. The four colour theorem

  11. Longitude1
    • Huygens and Auzout had been working on grinding lens and mirrors and had developed new telescopes which had enabled Huygens to compute Saturn's rotation period and to have discovered Saturn's ring system and one of Saturn's moons.
      Go directly to this paragraph

  12. Fair book insert
    • The diameter of two concentric circles are 500 and 470 what is the area of the ring.

  13. Maxwell's House
    • From this time forward I became very intimate with him, and we discussed together, with schoolboy enthusiasm, numerous schoolboy problems, among which I remember particularly the various plane sections of a ring or tore, and the form of a cylindrical mirror which should show one his own image unperverted.

  14. Tait's scrapbook
    • From this time forward I became very intimate with him, and we discussed together, with schoolboy enthusiasm, numerous schoolboy problems, among which I remember particularly the various plane sections of a ring or tore, and the form of a cylindrical mirror which should show one his own image unperverted.

  15. Bourbaki 1
    • The last section outlines an interesting method of treating structures, such as order, topology, group, ring, etc., on a general basis and having concepts like isomorphism defined quite generally.

Famous Curves

No matches from this section

Societies etc

  1. Minutes for 1989
    • R R Laxton (2) Nottingham Ring Theory .
    • J D P Meldrum EMS Ring Theory .

  2. BMC 1995
    • Special session: Ring theoryn Organiser: K A Brown .

  3. BMC 1977
    • Passmann, D SThe Jacobson radical of a group ring of a locally solvable group .

  4. Minutes for 1990
    • R R Laxton (4) Nottingham Ring Theory .

  5. Minutes for 2004
    • Ring Theory was mentioned as a topic which has not been treated in special sessions since 1995.

  6. European Mathematical Society Prizes
    • In further works, Seidel constructed a natural representation of the fundamental group of the group of Hamiltonian symplectomorphisms into the quantum cohomology ring.

References

  1. References for Amitsur
    • L H Rowen, Amitsur and ring theory, in A Mann, A Regev, L Rowen, D J Saltman and L W Small, Selected papers of S A Amitsur with commentary (2 Vols.) (American Mathematical Society, Providence, RI, 2001), 3-9.

  2. References for Herschel Caroline
    • F J Ring, John Herschel and his heritage, in D G King-Hele (ed.), John Herschel 1792-1871 : A bicentennial commemoration (London, 1992), 3-16.

  3. References for Liu Hui
    • S S Bai, A re-examination of a ring area problem in the 'Jiu zhang suanshu' (Chinese), Beijing Shifan Daxue Xuebao 30 (1) (1994), 139-142.

Additional material

  1. James Clerk Maxwell on the nature of Saturn's rings
    • We know, since it has been demonstrated by Laplace, that a uniform solid ring cannot revolve permanently about a planet.
    • We shall find that the stability of the motion of the ring would be ensured by loading the ring at one point with a heavy satellite about four and one-half times the weight of the ring, but this load, besides being inconsistent with the observed appearance of the rings, must be far too artificially adjusted to agree with the natural arrangements observed elsewhere, for a very small error in excess or defect would render the ring again unstable.
    • We are, therefore, constrained to abandon the theory of a solid ring, and to consider the case of a ring, the parts of which are not rigidly connected, as in the case of a ring of independent satellites, or a fluid ring.
    • There is now no danger of the whole ring or any part of it being precipitated on the body of the planet.
    • Every particle of the ring is now to be regarded as a satellite of Saturn, disturbed by the attraction of a ring of satellites at the same mean distance from the planet, each of which however is subject to slight displacements.
    • The mutual action of the parts of the ring will be so small compared with the attraction of the planet, that no part of the ring can ever cease to move round Saturn as a satellite.
    • But the question now before us is altogether different from that relating to the solid ring.
    • We have now to take account of variations in the form and arrangement of the parts of the ring, as well as its motion as a whole, and we have as yet no security that these variations may not accumulate till the ring entirely loses its original form, and collapses into one or more satellites, circulating round Saturn.
    • In this essay I have shown that such a destructive tendency actually exists, but that by the revolution of the ring it is converted into the condition of dynamical stability.
    • We found that the stability of the motion of a solid ring depended on so delicate an adjustment, and at the same time so unsymmetrical a distribution of mass, that even if the exact condition were fulfilled, it could scarcely last long, and if it did, the immense preponderance of one side of the ring would be easily observed, contrary to experience.
    • We next examined the motion of a ring of equal satellites, and found that if the mass of the planet is sufficient, any disturbances produced in the arrangement of the ring will be propagated round it in the form of waves, and will not introduce dangerous confusion.
    • If the satellites are unequal, the propagation of the waves will no longer be regular, but disturbances of the ring will in this, as in the former case, produce only waves, and not growing confusion.
    • Supposing the ring to consist, not of a single row of large satellites, but of a cloud of evenly distributed unconnected particles, we found that such a cloud must have a very small density in order to be permanent, and that this is inconsistent with its outer and inner parts moving with the same angular velocity.
    • Supposing the ring to be fluid and continuous, we found that it will be necessarily broken up into small portions.
    • We are not able to ascertain by observation the constitution of the two outer divisions of the system of rings, but the inner ring is certainly transparent, for the limb of Saturn has been observed through it.
    • It is also certain, that though the space occupied by the ring is transparent, it is not through the material parts of it that Saturn was seen, for his limb was observed without distortion; which shows that there was no refraction, and, therefore, that the rays did not pass through a medium at all, but between the solid or liquid particles of which the ring is composed.
    • Finally, the two outer rings have been observed for 200 years, and it appears, from the careful analysis of all the observations by Struve, that the second ring is broader than when first observed, and that its inner edge is nearer the planet than formerly.
    • The inner ring also is suspected to be approaching the planet ever since its discovery in 1850.
    • These appearances seem to indicate the same slow progress of the rings towards separation which we found to he the result of theory, and the remark, that the inner edge of the inner ring is most distinct, seems to indicate that the approach towards the planet is less rapid near the edge, as we had reason to conjecture.
    • As to the apparent unchangeableness of the exterior diameter of the outer ring, we must remember that the outer rings are certainly far more dense than the inner one, and that a small change in the outer rings must balance a great change in the inner one.

  2. James Clerk Maxwell on the nature of Saturn's rings
    • We know, since it has been demonstrated by Laplace, that a uniform solid ring cannot revolve permanently about a planet.
    • We shall find that the stability of the motion of the ring would be ensured by loading the ring at one point with a heavy satellite about four and one-half times the weight of the ring, but this load, besides being inconsistent with the observed appearance of the rings, must be far too artificially adjusted to agree with the natural arrangements observed elsewhere, for a very small error in excess or defect would render the ring again unstable.
    • We are, therefore, constrained to abandon the theory of a solid ring, and to consider the case of a ring, the parts of which are not rigidly connected, as in the case of a ring of independent satellites, or a fluid ring.
    • There is now no danger of the whole ring or any part of it being precipitated on the body of the planet.
    • Every particle of the ring is now to be regarded as a satellite of Saturn, disturbed by the attraction of a ring of satellites at the same mean distance from the planet, each of which however is subject to slight displacements.
    • The mutual action of the parts of the ring will be so small compared with the attraction of the planet, that no part of the ring can ever cease to move round Saturn as a satellite.
    • But the question now before us is altogether different from that relating to the solid ring.
    • We have now to take account of variations in the form and arrangement of the parts of the ring, as well as its motion as a whole, and we have as yet no security that these variations may not accumulate till the ring entirely loses its original form, and collapses into one or more satellites, circulating round Saturn.
    • In this essay I have shown that such a destructive tendency actually exists, but that by the revolution of the ring it is converted into the condition of dynamical stability.
    • We found that the stability of the motion of a solid ring depended on so delicate an adjustment, and at the same time so unsymmetrical a distribution of mass, that even if the exact condition were fulfilled, it could scarcely last long, and if it did, the immense preponderance of one side of the ring would be easily observed, contrary to experience.
    • We next examined the motion of a ring of equal satellites, and found that if the mass of the planet is sufficient, any disturbances produced in the arrangement of the ring will be propagated round it in the form of waves, and will not introduce dangerous confusion.
    • If the satellites are unequal, the propagation of the waves will no longer be regular, but disturbances of the ring will in this, as in the former case, produce only waves, and not growing confusion.
    • Supposing the ring to consist, not of a single row of large satellites, but of a cloud of evenly distributed unconnected particles, we found that such a cloud must have a very small density in order to be permanent, and that this is inconsistent with its outer and inner parts moving with the same angular velocity.
    • Supposing the ring to be fluid and continuous, we found that it will be necessarily broken up into small portions.
    • We are not able to ascertain by observation the constitution of the two outer divisions of the system of rings, but the inner ring is certainly transparent, for the limb of Saturn has been observed through it.
    • It is also certain, that though the space occupied by the ring is transparent, it is not through the material parts of it that Saturn was seen, for his limb was observed without distortion; which shows that there was no refraction, and, therefore, that the rays did not pass through a medium at all, but between the solid or liquid particles of which the ring is composed.
    • Finally, the two outer rings have been observed for 200 years, and it appears, from the careful analysis of all the observations by Struve, that the second ring is broader than when first observed, and that its inner edge is nearer the planet than formerly.
    • The inner ring also is suspected to be approaching the planet ever since its discovery in 1850.
    • These appearances seem to indicate the same slow progress of the rings towards separation which we found to he the result of theory, and the remark, that the inner edge of the inner ring is most distinct, seems to indicate that the approach towards the planet is less rapid near the edge, as we had reason to conjecture.
    • As to the apparent unchangeableness of the exterior diameter of the outer ring, we must remember that the outer rings are certainly far more dense than the inner one, and that a small change in the outer rings must balance a great change in the inner one.

  3. Cassini and the Division in Saturn's Ring
    • Cassini and the Division in Saturn's Ring .
    • In 1730 Giovanni Domenico Cassini wrote a paper The Discovery of the Division in Saturn's Ring which appeared in Volume X of the Memoires de I'Academie Royale des Sciences.
    • The Discovery of the Division in Saturn's Ring .
    • After the discoveries which have been made at different times concerning the globe of Saturn, its ring and its satellites, in part by Huygens who discovered one of the satellites which revolves around Saturn in 16 days less 47 minutes, and in part by Cassini who discovered two others of which we will give the history at an early date, it seemed that there was nothing more to discover concerning the planet; however, the latest observations that Cassini has made concerning the body of Saturn and its ring, show that in the Heavens as well as on the Earth, something new to observe always appears.
    • After the emergence of Saturn from the rays of the Sun as a morning star in the year 1675, the globe of the planet appeared with a dark band, similar to those of Jupiter, extending the length of the ring from East to West, as it is nearly always shown by the 34-foot telescope, and the breadth of the ring was divided by a dark line into two equal parts, of which the interior and nearer one to the globe was very bright, and the exterior part slightly dark.
    • This appearance gave an impression of a double ring, of which the inferior ring, being larger and darker, had superposed upon it another that is narrower and brighter, and reminds one that in the year 1671, when the extensions of Saturn were on the verge of disappearing they contracted beforehand, perhaps because the outer part of the ring, which was single and dark, disappeared before the inner part, which was double and brighter.
    • In the same year, 1671, the shorter diameter of the ring was still less than the diameter of the globe which extended outside the ring on the North and South sides, and this phase lasted until the immersion of Saturn in the rays of the Sun in the year 1676.
    • But after its emersion, which took place last summer, the shorter diameter of the ring exceeded that of the globe.
    • There is an observation by Hevelius in the English Journal, which corresponds to the first of these two phases; but as he has noted neither the band of Saturn, nor the distinction which can he seen in the ring, one has reason to judge that the telescopes which he uses are much inferior to those of the Royal Observatory.

  4. Cassini and the Division in Saturn's Ring
    • Cassini and the Division in Saturn's Ring .
    • In 1730 Giovanni Domenico Cassini wrote a paper The Discovery of the Division in Saturn's Ring which appeared in Volume X of the Memoires de I'Academie Royale des Sciences.
    • The Discovery of the Division in Saturn's Ring .
    • After the discoveries which have been made at different times concerning the globe of Saturn, its ring and its satellites, in part by Huygens who discovered one of the satellites which revolves around Saturn in 16 days less 47 minutes, and in part by Cassini who discovered two others of which we will give the history at an early date, it seemed that there was nothing more to discover concerning the planet; however, the latest observations that Cassini has made concerning the body of Saturn and its ring, show that in the Heavens as well as on the Earth, something new to observe always appears.
    • After the emergence of Saturn from the rays of the Sun as a morning star in the year 1675, the globe of the planet appeared with a dark band, similar to those of Jupiter, extending the length of the ring from East to West, as it is nearly always shown by the 34-foot telescope, and the breadth of the ring was divided by a dark line into two equal parts, of which the interior and nearer one to the globe was very bright, and the exterior part slightly dark.
    • This appearance gave an impression of a double ring, of which the inferior ring, being larger and darker, had superposed upon it another that is narrower and brighter, and reminds one that in the year 1671, when the extensions of Saturn were on the verge of disappearing they contracted beforehand, perhaps because the outer part of the ring, which was single and dark, disappeared before the inner part, which was double and brighter.
    • In the same year, 1671, the shorter diameter of the ring was still less than the diameter of the globe which extended outside the ring on the North and South sides, and this phase lasted until the immersion of Saturn in the rays of the Sun in the year 1676.
    • But after its emersion, which took place last summer, the shorter diameter of the ring exceeded that of the globe.
    • There is an observation by Hevelius in the English Journal, which corresponds to the first of these two phases; but as he has noted neither the band of Saturn, nor the distinction which can he seen in the ring, one has reason to judge that the telescopes which he uses are much inferior to those of the Royal Observatory.

  5. Christiaan Huygens' article on Saturn's Ring
    • Christiaan Huygens' article on Saturn's Ring .
    • In 1659 Christiaan Huygens published an article on Saturn's Ring in Systema Saturnium.
    • When, therefore, the planet continued day after day to present this same aspect, I came to understand that, inasmuch as the circuit of Saturn and the adhering bodies was so short, this could happen under no other condition than that the globe of Saturn were assumed to be surrounded equally on all sides by another body, and that thus a kind of ring encircled it about the middle; for so, with whatever velocity it revolved, it would always present the same aspect to us, if, of course, its axis were perpendicular to the plane of the ring.
    • Therefore, after that, I began to consider whether the other phases that Saturn was said to have could be accounted for by the same ring.
    • either side these arms projected did not follow the line of the ecliptic, but cut it at an angle of more than 20 degrees, I concluded that in the same way the plane of the ring which I had imagined was inclined at about the same angle to the plane of the ecliptic - with a permanent and unchanging inclination, be it understood, as is known to be the case on this Earth of ours with the plane of the equator.
    • From this inclination it necessarily followed that in its different aspects the same ring showed to us at one time a rather broad ellipse, at another time a narrower ellipse, and sometimes even a straight line.
    • As regards the handle-like formations, I understand that this phenomenon was due to the fact that the ring was not attached to the globe of Saturn, but was separated from it the same distance all around.
    • These facts, accordingly, being thus brought into line, and the above-mentioned inclination of the ring being also assumed, all the wonderful appearance of Saturn, I found, could be referred to this source, as will presently be shown.
    • ["It is encircled by a ring, thin, plane, nowhere attached, inclined to the ecliptic."] That the width of the space intervening between the ring and the globe of Saturn is equal to the width of the ring itself or even exceeds it, is shown by the figure of Saturn as observed by others.
    • and then more definitely by its figure as seen by myself; that, likewise, the ratio of the greatest diameter of the ring to the diameter of Saturn is about 9 to 4.
    • I believe that I should digress here to meet the objection of those who will find it exceedingly strange and possibly unreasonable that I should assign to one of the celestial bodies a figure the like of which has up to this time not been found in any one of them, although, on the other hand, it has been believed as certain, and considered as established by natural law, that the spherical form is the only one adapted to them; and that I should place this solid and permanent ring (for such I consider it) about Saturn, without attaching it by any joints or ties, although imagining that it preserves a uniform distance on every side and revolves in company with Saturn at a very high rate of speed.
    • These men should consider that I do not construct this hypothesis from pure invention and out of my own fancy, as the astronomers do their epicycles, which nowhere appear in the heavens, but that I perceive this ring very plainly with the eyes; with which, obviously, we discern the figures of all other things.
    • Furthermore, since, owing to the great similarity and relationship that exists between Saturn and our Earth, it seems possible to conclude quite conclusively that the former, like the latter, is situated in the middle of its own vortex, and that its centre has a natural tendency to reach toward all that is considered to have weight there, it must also result that the ring in question, pressing with all its parts and with equal force toward the centre, comes by this very fact to a permanent position in such a way that it is equally distant on all sides from that centre.

  6. Christiaan Huygens' article on Saturn's Ring
    • Christiaan Huygens' article on Saturn's Ring .
    • In 1659 Christiaan Huygens published an article on Saturn's Ring in Systema Saturnium.
    • When, therefore, the planet continued day after day to present this same aspect, I came to understand that, inasmuch as the circuit of Saturn and the adhering bodies was so short, this could happen under no other condition than that the globe of Saturn were assumed to be surrounded equally on all sides by another body, and that thus a kind of ring encircled it about the middle; for so, with whatever velocity it revolved, it would always present the same aspect to us, if, of course, its axis were perpendicular to the plane of the ring.
    • Therefore, after that, I began to consider whether the other phases that Saturn was said to have could be accounted for by the same ring.
    • either side these arms projected did not follow the line of the ecliptic, but cut it at an angle of more than 20 degrees, I concluded that in the same way the plane of the ring which I had imagined was inclined at about the same angle to the plane of the ecliptic - with a permanent and unchanging inclination, be it understood, as is known to be the case on this Earth of ours with the plane of the equator.
    • From this inclination it necessarily followed that in its different aspects the same ring showed to us at one time a rather broad ellipse, at another time a narrower ellipse, and sometimes even a straight line.
    • As regards the handle-like formations, I understand that this phenomenon was due to the fact that the ring was not attached to the globe of Saturn, but was separated from it the same distance all around.
    • These facts, accordingly, being thus brought into line, and the above-mentioned inclination of the ring being also assumed, all the wonderful appearance of Saturn, I found, could be referred to this source, as will presently be shown.
    • ["It is encircled by a ring, thin, plane, nowhere attached, inclined to the ecliptic."] That the width of the space intervening between the ring and the globe of Saturn is equal to the width of the ring itself or even exceeds it, is shown by the figure of Saturn as observed by others.
    • and then more definitely by its figure as seen by myself; that, likewise, the ratio of the greatest diameter of the ring to the diameter of Saturn is about 9 to 4.
    • I believe that I should digress here to meet the objection of those who will find it exceedingly strange and possibly unreasonable that I should assign to one of the celestial bodies a figure the like of which has up to this time not been found in any one of them, although, on the other hand, it has been believed as certain, and considered as established by natural law, that the spherical form is the only one adapted to them; and that I should place this solid and permanent ring (for such I consider it) about Saturn, without attaching it by any joints or ties, although imagining that it preserves a uniform distance on every side and revolves in company with Saturn at a very high rate of speed.
    • These men should consider that I do not construct this hypothesis from pure invention and out of my own fancy, as the astronomers do their epicycles, which nowhere appear in the heavens, but that I perceive this ring very plainly with the eyes; with which, obviously, we discern the figures of all other things.
    • Furthermore, since, owing to the great similarity and relationship that exists between Saturn and our Earth, it seems possible to conclude quite conclusively that the former, like the latter, is situated in the middle of its own vortex, and that its centre has a natural tendency to reach toward all that is considered to have weight there, it must also result that the ring in question, pressing with all its parts and with equal force toward the centre, comes by this very fact to a permanent position in such a way that it is equally distant on all sides from that centre.

  7. Jacobson: 'Structure of Rings
    • An important event in the development of ring theory was the publication of Structure of Rings by Nathan Jacobson in 1956.
    • For example, the theory of centralizers of finite dimensional simple subalgebras of simple rings with minimum condition appears as a special case of the Galois theory of the complete ring of linear transformations of a vector space over a division ring.
    • The starting point in our considerations is the definition of a radical for an arbitrary ring.
    • This is an ideal which measures the departure of a ring from semi-simplicity.
    • A semi-simple ring is one which has enough irreducible representations to distinguish elements.
    • A ring which has a faithful irreducible representation is called primitive.
    • The last part of this chapter deals with the Galois theory of the complete ring of linear transformations of a vector space over a division ring.
    • One of these is the lower nil radical of Baer which coincides with the intersection of the prime ideals of a ring.
    • In Chapter IX we define a topology of the set of primitive ideals of a ring and we use this to obtain representations of rings as rings of continuous functions on topological spaces.
    • The only knowledge assumed is that of the rudiments of ring and module theory such as is found in any of the introductory texts to abstract algebra.

  8. P G Tait's obituary of Listing
    • From this follow the properties of a large class of knots which form "clear coils." A special example of these, given by Listing for threads, is the well-known juggler's trick of slitting a ring-formed band up the middle, through its whole length, so that instead of separating into two parts, it remains in a continuous ring.
    • If three half-twists be given, the paper still remains a continuous band after slitting, but it cannot be opened into a ring, it is in fact a trefoil knot.
    • The reader, who wishes to have an elementary notion of the higher forms of problems treated by Listing, is advised to investigate the modification which Euler's formula would undergo if the polyhedron were (on the whole) ring-shaped: - as, for instance, an anchor-ring, or a plane slice of a thick cylindrical tube.

  9. Gordon Preston on semigroups
    • He then went off to the States for a year and my new supervisor was E C (Edward) Thompson and, with him, I changed my subject to algebraic geometry, with a strong emphasis on commutative ring theory and with not so much geometry.
    • I assumed that, on going down from Cambridge, David Rees had read this book of Albert's and tried, what was a natural thing to do, to reproduce the ring structure theory for a system without the addition operation.
    • Perhaps this ring theoretic notation was common in certain circles those days.

  10. André Weil: 'Algebraic Geometry
    • for instance, one will find here all that is needed for the proof of Bertini's theorems, for a detailed ideal-theoretic study (by geometric means) of the quotient-ring of a simple point, for the elementary part of the theory of linear series, and for a rigorous definition of the various concepts of equivalence.
    • formal power-series, and the representation of an ideal in a Noetherian ring as intersection of primary ideals) are used; the reader who is willing to take that theorem for granted, or successful in constructing a simpler proof of it, will not require, in all the rest of the book, any knowledge of these methods, or of anything beyond what has been mentioned above.
    • 1-85], for some of the main results in the theory of intersections, alternative proofs which begin by establishing the corresponding theorems for algebroid varieties, has shown how the ring of formal power-series can be given the principal role, instead of the subordinate one which it plays in our treatment.

  11. Jacobson: 'Theory of Rings
    • An important event in the development of ring theory was the publication of The Theory of Rings by Nathan Jacobson in 1943.
    • Moreover, the theory of modules, and hence representation theory, may be regarded as the study of a set of rings of endomorphisms all of which are homomorphic images of a fixed ring R.

  12. Malcev: 'Foundations of Linear Algebra' Introduction
    • In the present century linear algebra has acquired new richness and versatility through the use of the concepts of group and non-commutative ring in algebra itself, and through the use of in finite -dimensional function spaces in analysis.

  13. University of Glasgow Examinations
    • Find the temperature at any point of a thin solid ring, and at any time, supposing the initial temperature to have been uniformly of a constant value c through one half and the other half.

  14. Airy on Thales' eclipse
    • Thus it appears that an error of two seconds in Bradley's observations, (the angle which a finger-ring subtends at the distance of a mile, and which is smaller than can be perceived by the unassisted eye) would destroy our conclusions with regard to the distant eclipses in question.

  15. What do mathematicians do?
    • If a non-mathematician listens to these people talk or attempts to read their journals, he confronts an incomprehensible jargon filled with words like differential equation, group, ring, manifold, homotopy, etc.

  16. Kurosh: 'The theory of groups' 2nd edition
    • In addition, I might add that he should be acquainted with the concept of a ring and the simplest concepts connected with it.

  17. Semple and Kneebone: 'Algebraic Projective Geometry
    • Such a system is known in algebra as a ring.

  18. Gruenberg: 'Relation Modules
    • On the other hand, I hoped to persuade ring theorists that here was an area of group theory well suited to applications of integral representation theory.

  19. EMS obituary
    • It is easy to visualise the dissection, starting with p = 1 when R can be a torus, or anchor-ring, generated by the revolution of a circle round a line in its plane that does not meet it.

  20. The St Andrews Schmidt-Cassegrain Telescope
    • These pads can be seen through the grid, in the photograph of the lower end of the pilot model which is reproduced here; the ring of cylinders on the outer edge is also associated with this balancing system.

Quotations

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Chronology

  1. Mathematical Chronology
    • This leads to the development of ring theory.
    • Lasker proves the decomposition theorem for ideals into primary ideals in a polynomial ring.
    • Quillen formulates higher algebraic K-theory, a new tool that uses geometric and topological methods and ideas to formulate and solve major problems in algebra, particularly ring theory and module theory.

  2. Chronology for 1900 to 1910
    • Lasker proves the decomposition theorem for ideals into primary ideals in a polynomial ring.

  3. Chronology for 1840 to 1850
    • This leads to the development of ring theory.

  4. Chronology for 1970 to 1980
    • Quillen formulates higher algebraic K-theory, a new tool that uses geometric and topological methods and ideas to formulate and solve major problems in algebra, particularly ring theory and module theory.

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JOC/BS August 2001