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Paul Moritz Cohn

Born: 8 January 1924 in Hamburg, Germany
Died: 20 April 2006 in London, England

Paul Cohn studied at Trinity College Cambridge, and he was awarded a B.A. in 1948. He continued to study at Cambridge for his doctorate and this was awarded in 1951. In this same year he was appointed Charge de Recherches at the University of Nancy in France where he remained for a year.

In 1952 Cohn was appointed as a lecturer in mathematics at Manchester University. Cohn was a visiting professor at Yale University during 1961-62, spending part of 1962 at the University of California at Berkeley. This was the year that Cohn left his lectureship at Manchester to take up a Readership at Queen Mary College of the University of London. Cohn remained at Queen Mary College until 1967 but he spent some time on visiting appointments during those five years, holding visiting professorships at the University of Chicago in 1964 and at the State University of New York at Stony Brook in 1967.

Remaining within the University of London, Cohn moved to Bedford College in 1967 where he was appointed professor of mathematics and head of the Department of Mathematics. He was soon on his travels again, being a visiting professor at Rutgers University in 1967-68, at the University of Paris in 1969, and at Tulane University and the Indian Institute of Technology in Delhi both in 1971. Then in 1972 he was visiting professor at the University of Alberta and the following year he visited Carleton University in Ottawa. A visit to Israel in 1975 took him to Technion in Haifa and then in 1978 he was back in the United States at Iowa State University. The following year his travels took him back to Germany, the country of his birth, where he visited the University of Bielefeld.

Cohn moved again within the University of London in 1984 when he was appointed as professor at University College London. Two years later he was honoured with the title of Astor Professor of Mathematics at University College. He retired in 1989 being appointed Professor Emeritus and Honorary Research Fellow. In this period 1984-89, Cohn fitted in a number of further visits including a return visit to the University of Alberta in 1986 and a return visit to Israel in 1987, this time to visit Bar Ilan University at Ramat Gan.

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. His first papers appeared in print in 1952 and these early papers cover many topics. He generalised a theorem due to Magnus, and worked on the structure of tensor spaces. In 1953 he published a joint paper with K Mahler on pseudo-valuations and the following year he published a work on Lie algebras. Over the next few years his work ranged across group theory, field theory, Lie rings, semigroups, abelian groups and ring theory. His first book Lie groups was published in 1957.

From 1958 he published papers on Jordan algebras, Lie division rings, skew fields, free ideal rings and non-commutative unique factorisation domains. His second book Linear equations was published in 1958 and another book Solid geometry was published in 1961. A further book Universal algebra was published in 1965 with a second edition appearing in 1981. From the mid 1960s his work concentrates on non-commutative ring theory and the theory of algebras.

Perhaps Cohn's best known research monograph Free rings and their relations was published in 1971. This contained a systematic development of the work of Cohn and others on free associative algebras and related classes of rings, in particular free ideal rings. Contained in the book are Cohn's beautiful results on the embedding of rings into skew fields which he had published in papers earlier. A reviewer commented:-

On the whole, the book is a notable event in the literature of modern algebra. It completes the formation of the theory of free associative algebras and related classes of rings as an independent domain of ring theory.
A second edition of this book appeared in 1985 with additional material. A reviewer commented:-
This is the second edition of a book proven to be rather important in developing the subject of free (associative) algebras. Its importance is not only as a source for learning and reference but also as a collection of attractive open questions.
In 1974 the first volume of his undergraduate book Algebra was published. Volume II of Algebra appeared in 1977 and when the work appeared in a second edition it was a three volume work, with volumes I and II in 1982 and volume III published in 1990. Other books by Cohn include Skew field constructions (1977), Algebraic numbers and algebraic functions (1991), Elements of linear algebra (1994) and Skew fields published as Volume 57 in the Encyclopedia of Mathematics and its Applications. This book extends the lecture notes Skew field constructions published in 1977 but the 1995 work provides a comprehensive look at the whole area:-
The theory of skew fields is still not so familiar as the commutative analogue. The complexity of the problems in the noncommutative setting is one of the reasons for this fact. It is Cohn's merit to provide a coherent treatment of this subject which at the same time leads the reader to a wide range of interesting and important research problems, related to questions in algebra, geometry and logic.
Cohn was an enthusiastic member of the London Mathematical Society and he has served the Society as its secretary during 1965-67, as a Council member in 1968-71, 1972-75 and 1979-84, being President of the Society during 1982-84. He also acted as editor of the London Mathematical Society Monographs during 1968-77 and again 1980-93. He also served as a member of the Mathematical Committee of the Science Research Council from 1977 to 1980 and he served on the Council of the Royal Society of London in 1985-87. Cohn was elected a fellow of the Royal Society in 1980 and has received many honours for his outstanding contribution to mathematics. Among the various awards to Cohn have been the Lester R Ford Award from the Mathematical Association of America in 1972 and the Senior Berwick Prize of the London Mathematical Society in 1974.

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

September 1998
MacTutor History of Mathematics
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