**Alfred Goldie**'s father [1]:-

... worked as a skilled fitter at Austin Motors, the car manufacturers. The factory employed1 500skilled workers serving15 000unskilled labourers. The former were responsible for preparing the brass templates used for the accurate positioning of the holes for the screws that held together the various parts of the car. Templates were prepared for new models every two years, and had to be accurate to one thousandth of an inch. The skilled workers often had little work to do, as the actual drilling was done by the unskilled employees.

Alfred attended Wolverhampton Grammar School where he won a scholarship to St John's College, Cambridge. He entered Cambridge in 1939 just days after World War II began. His scholarship allowed him to complete the usual three year degree in only two years and, indeed, he obtained a First in Part II of the mathematical tripos in 1941. During these two years at Cambridge he was in the Officers Training Corps, and he also took an interest in politics taking a left wing Communist position.

Because of the war, Goldie did not continue to Part III of the tripos but instead was interviewed for a military position by C P Snow, the novelist, who was acting as scientific adviser to the British Government. Snow realised that, despite Goldie's training in the Officers Training Corps, he would make a more valuable contribution to the war effort using his mathematical skills in Ballistic Research. He was sent to work under C A Clemmow in Cambridge, but soon the Ballistic Research team moved to Shrewsbury. He also spent time at a military base near Glasgow, then towards the end of the war he was sent to the Woolwich Arsenal in London. In October 1944 he married Mary Kenyon; they had one son and two daughters. As the war drew to a close he began to think about restarting his mathematical education [1]:-

[

He]met G E H Reuter, who was at the time a research student working in Cambridge under Hardy and Littlewood. They discussed the possibility of Goldie's going back to Cambridge to finish his education by doing Part III of the Mathematical Tripos. However ... Reuter warned him that as a married man Goldie would not be able to afford another year as a student.

Goldie requested that he be allowed to begin studies for a Ph.D. at London University, and he was allowed to take some time off from his work at Woolwich Arsenal to undertake mathematical research. As part of his return to mathematics he began attending meetings of the London Mathematical Society. There he learnt that Philip Hall who, after undertaking work on codes at Bletchley Park, had returned to Cambridge. Since he wanted to undertake research in algebra, Goldie was advised to write to Hall seeking advice. This was readily given, and Hall suggested that Goldie read van der Waerden's *Moderne Algebra*, a difficult task for someone who did not speak German, but with the help of a dictionary Goldie soon mastered this fundamental text largely based on Emmy Noether's revolutionary contributions to algebra. Goldie exchanged letters with Hall, and also met Hall on some weekends when he was at his mother's home in Hampstead.

World War II ended in 1945 with Germany surrendering in May and Japan in August. Goldie continued to work at Woolwich Arsenal for another year, only leaving in September 1946 when he was appointed Assistant Lecturer in Mathematics at Nottingham University. At this stage he had not completed his Ph.D. at London University, but he decided that now he was in an academic position there was no need for him to continue with his doctoral studies. Alfred and Mary's first child, a son, was born a few weeks after they moved to Nottingham. He continued to write to Hall for advice regarding his research, and Hall suggested that he read the recently published first volume of *Algebra* by Bourbaki. This led Goldie to the results in universal algebra which he published in *The Jordan-Hölder theorem for general abstract algebras* (1950) and *The scope of the Jordan-Hölder theorem in abstract algebra* (1952).

A heavy lecture load and difficult living conditions for Goldie and his family, made his two years in Nottingham quite difficult. When he received an offer of a Lectureship at Newcastle University he did not hesitate but happily made the move. It made sense for several different reasons; he was able to afford much better housing for his family, and it was an exciting time there mathematically. Newcastle was building up a strong mathematics department guided by the professor, W Rogosinski. Kurt Hirsch was appointed to a Readership while Frank Bonsall and Alfred Goldie were appointed to Lectureships. 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. However, Bonsall was working on normed rings and this led to the two collaborating on the papers *Algebras which represent their linear functionals* (1953) and *Annihilator algebras* (1954). 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 [1]:-

One of the most characteristic features of ring theory in the late1940s and early1950s was the search for a structure theory for general rings. The most successful approach to this problem was that of N Jacobson. Thus it is not really surprising that Goldie's early work in ring theory was influenced by Jacobson's papers and books.

Jacobson and Goldie had met at the International Congress of Mathematicians in Amsterdam in 1954 and at that time Jacobson suggested that Goldie work on problems in Notherian rings. In fact Goldie's first paper in this area *Decompositions of semi-simple rings* (1956) made an immediate impact since Jacobson included one of Goldie's theorems in his classic monograph *Structure of Rings* of 1956, acknowledging that it had been communicated by Goldie. Over the next few years Goldie's work on non-commutative Notherian rings would totally revolutionise the subject. He was able to prove totally unexpected structure theorems. Even his first steps towards these results were startling as the following story, recounted in [1], tells us:-

After the war, whenever Brauer visited England he would go to Newcastle and stay with the Rogosinskis. On one of these occasions, Goldie showed Brauer his result on Noetherian domains. All Brauer said was: "Oh my God!" Goldie then explained that he now needed to "do the matrix case"; Brauer replied "my boy, you are on your own."

The breakthrough which allowed him to complete his structure theorem for Noetherian prime rings came after a family holiday to Sedberg in the Yorkshire Dales National Park was abandoned early due to persistent rain [1]:-

After five days they decided to give up, and returned to Newcastle on a Thursday. The break must have given Goldie's subconscious time to catch up, for between Friday and Monday he[made the breakthrough].

Goldie published his results, now known as "Goldie's Theorem," in *The structure of prime rings with maximum conditions* (1958) and *The structure of prime rings under ascending chain conditions* (1958). A generalisation appears in *Semi-prime rings with maximum condition* (1960). Robson explains in [2]:-

This result came as a huge surprise, since it had been unsuspected, and it was hailed as a most considerable advance. It led to invitations to work in other institutions, including Yale University, the Institut des Hautes Études Scientifiques in Paris and Tulane University in New Orleans. The developments since his discovery have only served to demonstrate just how seminal it was. It remains a major tool in the subject and in surrounding areas. It is pertinent to note that a1987graduate text which aimed to describe the result and to survey some of its consequences("Noncommutative Noetherian Rings" by J C McConnell and J C Robson)uses about50pages to establish Goldie's Theorem and over500pages to outline consequences.

Goldie had been promoted to Senior Lecturer at Newcastle, then in 1963 he was appointed Professor of Pure Mathematics at Leeds University. His mission at Leeds was to build a strong research group there. Indeed he was very successful in this and built a school of algebra at the University of Leeds that has had great influence on algebra for many years. Small writes [3]:-

At Leeds he continued his work in algebra, building a "school" in ring theory, hosting many visitors, and organizing memorable conferences. Goldie visited the U.S. many times, with extended stays at Yale, Tulane, and the University of California at San Diego.

He retired in 1986 but continued with his research, living in Bowness-on-Windermere. During his retirement he published papers such as (with Günter Krause) *Associated series and regular elements of Noetherian rings* (1987), *Rings with an additive rank function* (1990), and (with Günter Krause) *Embedding rings with Krull dimension in Artinian rings* (1996). His wife Mary died in 1995 and Goldie married Margaret Turner in 2002. His death was the result of a heart attack following surgery a few days earlier.

Among the honours Goldie received for his outstanding contributions, we mention the Senior Berwick Prize from the London Mathematical Society in 1970. He served on the Council of the London Mathematical Society and was Vice-President during 1978-80. He was Chairman of the British Mathematical Colloquium at Leeds in 1968, and lectured at the Colloquium of 1972 in Glasgow when he spoke on *Some infinite dimensional algebras*.

As to his character we quote from [2]:-

He was a vivid personality with an individual mind, full of opinions and strongly argued positions on mathematics but also on the wider world, including politics, where his views moved rightwards over the years - starting with Communism whilst a student at Cambridge, and ending well within traditional Conservatism. ... Alfred Goldie was a very practical man, particularly enjoying working with wood. He also had a love of the outdoors, which he shared with his first wife, Mary, a geographer. Despite somewhat incompatible personalities, they still managed to give their three children a stable and happy upbringing.

Also we quote from [3]:-

Alfred Goldie approached his work with optimism and with no fear of the really hard problems.

**Article by:** *J J O'Connor* and *E F Robertson*