# Paul Moritz Cohn

### Quick Info

Hamburg, Germany

London, England

**Paul Cohn**was a German-born mathematician who worked in England, mainly in the area of algebra, especially non-commutative rings.

### Biography

**Paul Cohn**was the only child of Jacob Cohn (1883-1942), known as James, and Julia Mathilde Cohen (1888-1941). Jacob Cohn was born in Hamburg on 10 September 1883, the son of Moritz Cohn and his wife Frumet Schwartz. Jacob, the only boy with four sisters, had to help support his family from the age of twelve after the death of his father. After he married Julia Mathilde on 18 February 1921, he became a partner in his father-in-law's cigar wholesale company Maass & Cohen; he became the owner in 1926. Julia Mathilde Cohn, born on 14 October 1888 in Hamburg, was the daughter of Ferdinand Siegmund Cohen and Rebecca Seeler. She was employed by the Hamburg School Authority and took up a position as a teacher at the school at Humboldtstrasse 30. When Jacob and Julia were married they lived on Isestrasse with Julia's widowed mother, Rebecca Cohen. They were living there when their son Paul Moritz Cohn was born but, after Julia's mother died in 1925, the family moved to a newly-built apartment at Lattenkamp 82.

The family were Jewish; Paul writes [22]:-

We identified as Jews, but we weren't religious.Jacob Cohn had fought in World War I, was wounded several times, and this led to him becoming a pacifist. The cigar company did not do well, partly because Jacob had ill health and partly because of the Depression, and by 1926 it seems to have essentially stopped trading. Paul's mother was working, so the family had sufficient to live on and employed a home help.

In 1928, when Paul was only four years old, he developed scarlet fever. Although today this is seldom a very serious disease, this was not so in the 1920s when many children died of the disease. Too ill to walk, he was taken to hospital in a horse and cart. His schooling began in April 1930 when he entered the Alsterdorfer Strasse School. At first he got on well at this school enjoying the academic work for which he was eager. He did suffer from being teased by his fellow pupils but he felt this was because he was a studious type; he did not feel it was antisemitism. Sadly things changed when his teacher became ill and another teacher took over his class. This new teacher kept picking on him and handing out punishments to him when he had not done anything wrong. Paul's parents spoke to the head teacher, complaining about Paul's treatment, and were told that this teacher was a National Socialist. The only solution was to change Paul to a different school so, in 1931, he moved to the Meerweinstrasse School where his mother was a teacher. She had been transferred from the Humboldtstrasse School to the newly-built Meerweinstrasse School in 1930. Paul got on well at this school for the following two and a half years.

On 30 January 1933 Hitler came to power and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from schools and universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. This had serious problems for the Cohns. Jacob Cohn's firm was shut down in 1933 (it was liquidated in 1938, and taken off the trade register on 6 January 1939) and in October 1933 Julia Cohn was dismissed from her teaching position [22]:-

After that she was unable to find steady work. She occasionally taught private lessons. She was granted a small pension, which she received from 1 November 1933 until 30 November 1941, after several school authorities who knew her from the Humboldtstrasse school pleaded her case with the State School Authority. The family lived from this pension and their savings.Paul's parents decided to send him to the Hamburg Jewish School. This had the disadvantage of giving him a long distance to travel. There was another problem which was that the Jewish school was much further ahead in the syllabus than the Meerweinstrasse School. It meant that, for the first time, Paul had to engage in intensive study to catch up. He found that he enjoyed intensive study! In 1934 he sat the entrance examinations for secondary school and did well. Secondary school went well and he had some excellent teachers.

You can read Paul Cohn's own description of his years in Hamburg, including details of his secondary schooling at THIS LINK.

After the Nazis came to power Jacob Cohn found it difficult to obtain employment and, between 1933 and 1938 [22]:-

... he worked at various companies as a bookkeeper. His last position was with J Jacobi & Co, an international shipping company. On 13 March 1938, the Hamburg Foreign Exchange Office issued a security order against the company's owner and revoked his authority to manage and represent the company. It was liquidated and the remaining assets were transferred to an "Aryan" businessman. Thereafter, Jacob Cohn found only temporary jobs, one of which was with the Jewish Community.In 1937 the Cohn family moved to Klosterallee which had the double advantage of being closer to Paul's school and also was a Jewish area where they felt more secure. On the Kristallnacht (so called because of the broken glass in the streets on the following morning), the 9-10 November 1938, 91 Jews were murdered, hundreds were seriously injured, and thousands were subjected to horrifying experiences. Thousands of Jewish businesses were burnt down together with over 150 synagogues. The Gestapo arrested 30,000 Jews and one of them was Jacob Cohn who was taken to the Sachsenhausen concentration camp. Paul Cohn's education was disrupted since most of his teachers were now in prison. In order to get Jacob Cohn released, Julia had to find a way for the family to emigrate. This, however, proved almost impossible since to emigrate they would need someone in the country they were planning to enter to provide a guarantee to look after them; how could they possibly find such a person. When the Netherlands offered to admit children without a guarantee, Paul's mother immediately registered him, and he began to learn Dutch. In March 1939 Jacob Cohn was released but told if his did not emigrate he would be arrested again. He feared that the Netherlands might not prove to be a safe place for his son in the longer term (he would be proved right) so they looked at another option, the KinderTransport which allowed children between the ages of 5 and 17 to reach Britain as refugees. The children were to travel in sealed trains. The first transport left barely six weeks after the Kristallnacht, the last, just two days before war broke out, which put an end to the programme. Paul Cohn waved goodbye to his parents when he boarded the train on 21 May 1939; he would never see his parents again.

Arriving in England [3]:-

... Paul was greeted at Liverpool Street Station in London by Mrs Lisbet Mueller-Hartmann, whom he remembered well as a distant relation from Hamburg. From there she escorted him by Underground to Victoria Station and arranged for him to take the train to Dorking, Surrey; there he was met by a lady from the refugee committee, who drove him to a farm at Newdigate, where Mr and Mrs Panning kept about 5000 chickens. Being over the school age of 14 years, Paul was required to work on the farm (unpaid, as this was a requirement of being accepted for Kindertransport).Cohn's work involved feeding and mucking out the chickens. It was fairly easy work but he had to work for 70 hours a week with only three afternoons off every fortnight. He was given a small amount of pocket money which let him go to the cinema where he often watched the same film several times, all in an attempt to improve his English. He was able to correspond with his parents using various tricks like sending the letter via friends in the United States, making sure there was no evidence that it originated in England [22]:-

Beginning on 19 September 1941, Jacob and Julia Cohn were required to wear the "Jews' star." Paul Cohn recalls that the last time he heard from his parents was in October 1941, through the Red Cross. He never saw them again. On orders of the Hamburg Gestapo, dated 4 December 1941, Jacob and Julia Cohn were deported to Riga on 6 December. They did not survive. Paul Cohn was prepared for the news when he learned of his parents' death after the war. Nothing is known about the exact circumstances of their deaths, and their dates of death were legally declared as 8 May 1945.At the end of 1941 the chicken farm closed since feed for the chickens was no longer available. Cohn then trained as an engineer and worked as a bench-fitter for four and a half years in a factory in London [3]:-

He was still in touch with the refugee committee in Dorking, which, recognising his intelligence and love of learning, and his special interest in mathematics, encouraged him to study for the Cambridge Entrance Examination and the School Certificate and Higher School Certificate Examinations, all of which he did by correspondence course. At that time Latin was still a requirement to enter the University of Cambridge, so he studied Latin from scratch. During his studies in his unheated room and before he started work, Paul needed to heat up his pen because the ink in it would freeze during the winter.Awarded an Exhibition to study mathematics at Trinity College, Cambridge, he began his study of the tripos in 1944. This was not the end of his problems, however, for after one term he was informed that his release from the factory had been an error and he was required to return. He had to work in the factory for a further year but, having only one term of study behind him, was able to pass the first year examinations. Returning to Trinity College, he was awarded a B.A. in 1948. He continued to study at Cambridge for his doctorate, advised by Philip Hall, and this was awarded in 1951 for his thesis

*Integral Modules, Lie Rings and Free Groups*. In this same year he was appointed Charge de Recherches at the University of Nancy in France where he remained for a year.

Cohn's first two papers were published in 1952:

*A theorem on the structure of tensor-spaces*and

*Generalization of a theorem of Magnus*. The first of these begins as follows:-

The present paper arose from an attempt to classify the submodules of a free associative algebra which admit all linear transformations of the free generators.The acknowledgements in these two papers are, respectively:-

I should like to thank Mr P Hall for his very helpful criticisms and suggestions. I am also indebted to the Department for Scientific and Industrial Research for a grant.In 1952 Cohn was appointed as a lecturer in mathematics at Manchester University. It was there he met the psychology undergraduate Deirdre Sonia Sharon, who was born in London of Jewish descent. Paul and Deirdre were married on 27 March 1958; they had two daughters, Susan Juliet Cohn (born in Manchester) and Ursula Yael Cohn (born in London).

I should like to thank Mr P Hall and Dr G Higman for their very helpful suggestions and encouragement, and the latter for allowing me to read his thesis. I am also indebted to the Department for Scientific and Industrial Research for a grant.

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 a 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.

For information about Cohn's books, see THIS LINK.

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 concentrated 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 earlier papers. The reviewer Leonid A Bokut 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. The reviewer Leonid G Makar-Limanov 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.For longer extracts from these two reviews, see THIS LINK.

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. We learn something about how another of his books

*Skew field constructions*(1977) came about from his Preface:-

As the name indicates, these really are lecture notes, though not for a single set of lectures. For this reason they may lack the polish of a book, but it is hoped that they have not entirely lost the directness of a lecture. The material comes from courses I have given in Manchester and London; some parts follow rather closely lectures given at Tulane University (1971), the University of Alberta (1972), Carleton University (1973), Tübingen (1974), Mons (1974), Haifa Technion (1975), Utrecht (1975) and Ghent (1976). It is a pleasure to acknowledge the hospitality of these institutions, and the stimulating effect of such critical audiences.Other books by Cohn include

*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 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.

Let us end with the 'Summary and Appreciation' from [3]:-

Paul Cohn's achievements in non-commutative ring theory are ones of which he could feel community, both at the justifiably proud. Moreover his books contributed greatly to algebraic knowledge in the mathematical research level and in undergraduate texts. He was greatly revered for these reasons. His quiet personality, coupled with the ability to listen and respond, was appreciated by students and other researchers alike. He was respected and admired the world over for these qualities. Paul Cohn gave great support to the London Mathematical Society and was its President from 1982 to 1984. Paul Cohn was a loving family man who never forgot his background in Germany and who always remembered his parents' sacrifice and devotion in sending him by Kindertransport to England. He cherished his UK citizenship, and England was his home. He had a great love of all activity, including walking in the Alps. Moreover he loved mathematics, to which he contributed greatly.

### References (show)

- T Barnard, Review: Introduction to ring theory, by Paul M Cohn,
*The Mathematical Gazette***85**(503) (2001), 362. - G M Bergman, Review: Skew field constructions (1977), by Paul M
*Cohn, Bull. Amer. Math. Soc.***1**(1979), 414-420. - G Bergman and T Stuart, Paul Moritz Cohn. 8 January 1924 - 20 April 2006,
*Biographical Memoirs of Fellows of the Royal Society***60**(2014), 127-150. - B Brainerd, Review: Universal algebra, by Paul M Cohn,
*Amer. Math. Monthly***74**(7) (1967), 878-880. - C R Butler, Review: Algebra Vols 1, 2 and 3 (Second paperback edition), by Paul M Cohn,
*The Mathematical Gazette***81**(491) (1997), 330-332. - R Curtis, Review: Further algebra and applications, by Paul M Cohn,
*The Mathematical Gazette***88**(512) (2004), 381. - A Czerniakiewicz, Review: Free rings and their relations, by Paul M Cohn,
*Bull. Amer. Math. Soc.***79**(1973), 873-878. - D J Fieldhouse, Review: Free rings and their relations, by Paul M Cohn,
*Amer. Math. Monthly***80**(5) (1973), 573. - H Freedman, Review: Algebra Volume 2 (Second Edition), by Paul M Cohn,
*The Mathematical Gazette***75**(471) (1991), 122. - H Freedman, Review: Algebra Volume 3 (Second Edition), by Paul M Cohn,
*The Mathematical Gazette***76**(477) (1992), 426-427. - J Gani, Obituary: Paul Cohn, FRS,
*Math. Sci.***31**(2) (2006), 131. - R L Goodstein, Review: Linear equations, by Paul M Cohn,
*The Mathematical Gazette***43**(344) (1959), 140-141. - L C Grove, Review: Basic algebra: groups, rings and fields, by Paul M Cohn,
*SIAM Review***45**(3) (2003), 607-608. - K A Hirsch, Review: Lie groups, by Paul M Cohn,
*The Mathematical Gazette***44**(347) (1960), 78-79. - G Leversha, Review: Basic algebra: groups, rings and fields, by Paul M Cohn,
*The Mathematical Gazette***89**(514) (2005), 153-154. - J Lewin, Review: Free rings and their relations (Second Edition), by Paul M Cohn,
*Bull. Amer. Math. Soc.***21**(1989), 139-142. - J D P Meldrum, Review: Algebra Volume 1, by Paul M Cohn,
*The Mathematical Gazette***59**(407) (1975), 53. - J D P Meldrum, Review: Algebra Volume 2, by Paul M Cohn,
*The Mathematical Gazette***62**(420) (1978), 137. - A Nijenhuis, Review: Lie groups, by Paul M Cohn,
*Bull. Amer. Math. Soc.***65**(1959), 338-341. - Paul M Cohn,
*London Mathematical Society Newsletter***350**(July 2006), 8. - P Quill, Review: Classic algebra, by Paul M Cohn,
*The Mathematical Gazette***86**(505) (2002), 175. - A Reinfeldt, Julia Cohn (née Cohen) born 1888,
*Stolpersteine Hamburg*(2014).

https://www.stolpersteine-hamburg.de/en.php?MAIN_ID=7&BIO_ID=3271 - A Robinson, Review: Universal algebra, by Paul M Cohn,
*The Journal of Symbolic Logic***34**(1) (1969), 113-114. - A Schofield, Professor Paul Cohn,
*The Independent*(8 August 2006). - H C Wang, Review: Lie groups, by Paul M Cohn,
*Amer. Math. Monthly***65**(8) (1958), 646.

### Additional Resources (show)

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Written by J J O'Connor and E F Robertson

Last Update March 2021

Last Update March 2021