Paul Cohn
Encyclopedia
Paul Moritz Cohn FRS (8 January 1924, Hamburg
, Germany
– 20 April 2006, London
, England
) was Astor Professor of Mathematics
at University College London
, 1986-9, and author of many textbooks on algebra
. His work was mostly in the area of algebra
, especially non-commutative rings
.
Both of his parents were born in Hamburg, as were three of his grandparents. His ancestors came from various parts of Germany. His father fought in the German army in World War I
; he was wounded several times and awarded the Iron Cross
. A street in Hamburg is named in memory of his mother.
When he was born, his parents were living with his mother's mother in Isestraße. After her death in October 1925, the family moved to a rented flat in a new building in Lattenkamp, in the Winterhude
quarter. He attended a kindergarten then, in April 1930, moved to Alsterdorfer Straße School. After a while, he had a new teacher, a National Socialist, who picked on him and punished him without cause. Thus in 1931 he moved to the Meerweinstraße School where his mother taught.
Following the rise of the Nazis in 1933, his father's business was confiscated and his mother dismissed. He moved to the Talmud-Tora-Schule, a Jewish school. In mid 1937, the family moved to Klosterallee. This was nearer the school, the synagogue and other pupils, being in the Jewish area. His German
teacher was Dr. Ernst Loewenberg, the son of the poet Jakob Loewenberg.
On the night of 9/10 November 1938 (Kristallnacht
), his father was arrested and sent to Sachsenhausen concentration camp
. He was released after four months, but told to emigrate. Cohn went to Britain in May 1939 on the Kindertransport
to work on a chicken farm, and never saw his parents again. He corresponded regularly with them until late 1941. At the end of the War
, he learned that they were deported to Riga
on 6 December 1941 and never returned. At the end of 1941 the farm closed. He trained as a precision engineer, acquired a work permit and worked in a factory for 4½ years. He passed the Cambridge Scholarship Examination, and won an exhibition to Trinity College, Cambridge
.
) in 1951. He then spent a year as a Chargé de Recherches at the University of Nancy. On his return, he became lecturer in mathematics at Manchester University. He was a visiting professor at Yale University
in 1961-62, and for part of 1962 was at the University of California at Berkeley. On his return, he became Reader at Queen Mary College
. He was a visiting professor at the University of Chicago
in 1964 and at the State University of New York at Stony Brook
in 1967. By then, he was regarded as one of the World's leading algebraists.
Also in 1967, he became head of the Department of Mathematics at Bedford College. He held several visiting professorships, in America, Paris
, Delhi
, Canada
, Haifa
and Bielefeld
. He was awarded 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.
In the early 1980s, funding cuts caused the closure of the small colleges of the University of London. Cohn moved to University College in 1984, together with the two other experts at Bedford on ring theory, Bill Stephenson and Warren Dicks. He became Astor Professor of Mathematics there in 1986. He continued to be a visiting professor, for example to the University of Alberta
in 1986 and to Bar Ilan University in 1987. He retired in 1989, but remained active as Professor Emeritus and Honorary Research Fellow until his death.
He was President of the London Mathematical Society, 1982-4, having been its secretary, 1965–67 and a Council member in 1968-71, 1972–75 and 1979-82. He was editor of the Society's Monographs in 1968-77 and 1980-93. He was elected a Fellow of the Royal Society
in 1980 and was on its council, 1985-87. He was a member of the Mathematical Committee of the Science Research Council, 1977-1980. He chaired the National Committee for Mathematics, 1988-9.
, and worked on the structure of tensor
spaces. In 1953 he published a joint paper with Kurt Mahler
on pseudo-valuations and in 1954 he published a work on Lie algebras.
Papers over the next few years covered areas such as group theory
, field theory
, Lie ring
s, semigroup
s, Abelian group
s and ring theory
. He published his first book, on Lie group
s, in 1957. After that, he moved into the areas of Jordan algebra
s, Lie division rings, skew fields, free ideal ring
s and non-commutative unique factorisation domains. He published his second book, Linear equation
s, in 1958 and his third, Solid geometry
, in 1961. Universal algebra appeared in 1965 (second edition 1981). After that, he concentrated on non-commutative ring theory and the theory of algebras.
His monograph Free rings and their relations appeared in 1971. It covered the work of Cohn and others on free associative algebras and related classes of rings, especially free ideal rings. He included all of his own published results on the embedding of rings into skew fields. The second, enlarged edition appeared in 1985.
Cohn also wrote undergraduate textbooks. Algebra volume I appeared in 1974 and volume II in 1977. The second edition, in three volumes, appeared in 1982–1990.
Hamburg
-History:The first historic name for the city was, according to Claudius Ptolemy's reports, Treva.But the city takes its modern name, Hamburg, from the first permanent building on the site, a castle whose construction was ordered by the Emperor Charlemagne in AD 808...
, Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
– 20 April 2006, London
London
London is the capital city of :England and the :United Kingdom, the largest metropolitan area in the United Kingdom, and the largest urban zone in the European Union by most measures. Located on the River Thames, London has been a major settlement for two millennia, its history going back to its...
, England
England
England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...
) was Astor Professor of Mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
at University College London
University College London
University College London is a public research university located in London, United Kingdom and the oldest and largest constituent college of the federal University of London...
, 1986-9, and author of many textbooks on algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
. His work was mostly in the area of algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, especially non-commutative rings
Noncommutative ring
In mathematics, more specifically modern algebra and ring theory, a noncommutative ring is a ring whose multiplication is not commutative; that is, if R is a noncommutative ring, there exists a and b in R with a·b ≠ b·a, and conversely.Noncommutative rings are ubiquitous in mathematics, and occur...
.
Ancestry and early life
He was the only child of Jewish parents, James (or Jakob) Cohn, owner of an import business, and Julia (née Cohen), a schoolteacher.Both of his parents were born in Hamburg, as were three of his grandparents. His ancestors came from various parts of Germany. His father fought in the German army in World War I
World War I
World War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...
; he was wounded several times and awarded the Iron Cross
Iron Cross
The Iron Cross is a cross symbol typically in black with a white or silver outline that originated after 1219 when the Kingdom of Jerusalem granted the Teutonic Order the right to combine the Teutonic Black Cross placed above a silver Cross of Jerusalem....
. A street in Hamburg is named in memory of his mother.
When he was born, his parents were living with his mother's mother in Isestraße. After her death in October 1925, the family moved to a rented flat in a new building in Lattenkamp, in the Winterhude
Winterhude
Winterhude is a quarter in the borough Hamburg-Nord of Hamburg, Germany. In 2007 the population was 48,799.-History:Winterhude was first mentioned in the 13th century, but archeological findings of tools, weapons and grave-mounds were dated in 1700 bC and 700 bC.During World War II the port of...
quarter. He attended a kindergarten then, in April 1930, moved to Alsterdorfer Straße School. After a while, he had a new teacher, a National Socialist, who picked on him and punished him without cause. Thus in 1931 he moved to the Meerweinstraße School where his mother taught.
Following the rise of the Nazis in 1933, his father's business was confiscated and his mother dismissed. He moved to the Talmud-Tora-Schule, a Jewish school. In mid 1937, the family moved to Klosterallee. This was nearer the school, the synagogue and other pupils, being in the Jewish area. His German
German language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....
teacher was Dr. Ernst Loewenberg, the son of the poet Jakob Loewenberg.
On the night of 9/10 November 1938 (Kristallnacht
Kristallnacht
Kristallnacht, also referred to as the Night of Broken Glass, and also Reichskristallnacht, Pogromnacht, and Novemberpogrome, was a pogrom or series of attacks against Jews throughout Nazi Germany and parts of Austria on 9–10 November 1938.Jewish homes were ransacked, as were shops, towns and...
), his father was arrested and sent to Sachsenhausen concentration camp
Sachsenhausen concentration camp
Sachsenhausen or Sachsenhausen-Oranienburg was a Nazi concentration camp in Oranienburg, Germany, used primarily for political prisoners from 1936 to the end of the Third Reich in May, 1945. After World War II, when Oranienburg was in the Soviet Occupation Zone, the structure was used as an NKVD...
. He was released after four months, but told to emigrate. Cohn went to Britain in May 1939 on the Kindertransport
Kindertransport
Kindertransport is the name given to the rescue mission that took place nine months prior to the outbreak of the Second World War. The United Kingdom took in nearly 10,000 predominantly Jewish children from Nazi Germany, Austria, Czechoslovakia, Poland and the Free City of Danzig...
to work on a chicken farm, and never saw his parents again. He corresponded regularly with them until late 1941. At the end of the War
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
, he learned that they were deported to Riga
Riga
Riga is the capital and largest city of Latvia. With 702,891 inhabitants Riga is the largest city of the Baltic states, one of the largest cities in Northern Europe and home to more than one third of Latvia's population. The city is an important seaport and a major industrial, commercial,...
on 6 December 1941 and never returned. At the end of 1941 the farm closed. He trained as a precision engineer, acquired a work permit and worked in a factory for 4½ years. He passed the Cambridge Scholarship Examination, and won an exhibition to Trinity College, Cambridge
Trinity College, Cambridge
Trinity College is a constituent college of the University of Cambridge. Trinity has more members than any other college in Cambridge or Oxford, with around 700 undergraduates, 430 graduates, and over 170 Fellows...
.
Career
He received a B.A in Mathematics from Cambridge University in 1948 and a Ph.D. (supervised by Philip HallPhilip Hall
Philip Hall FRS , was an English mathematician.His major work was on group theory, notably on finite groups and solvable groups.-Biography:...
) in 1951. He then spent a year as a Chargé de Recherches at the University of Nancy. On his return, he became lecturer in mathematics at Manchester University. He was a visiting professor at Yale University
Yale University
Yale University is a private, Ivy League university located in New Haven, Connecticut, United States. Founded in 1701 in the Colony of Connecticut, the university is the third-oldest institution of higher education in the United States...
in 1961-62, and for part of 1962 was at the University of California at Berkeley. On his return, he became Reader at Queen Mary College
Queen Mary, University of London
Queen Mary, University of London is a public research university located in London, United Kingdom and a constituent college of the federal University of London...
. He was a visiting professor at the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
in 1964 and at the State University of New York at Stony Brook
State University of New York at Stony Brook
The State University of New York at Stony Brook, also known as Stony Brook University, is a public research university located in Stony Brook, New York, on the North Shore of Long Island, about east of Manhattan....
in 1967. By then, he was regarded as one of the World's leading algebraists.
Also in 1967, he became head of the Department of Mathematics at Bedford College. He held several visiting professorships, in America, Paris
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...
, Delhi
Delhi
Delhi , officially National Capital Territory of Delhi , is the largest metropolis by area and the second-largest by population in India, next to Mumbai. It is the eighth largest metropolis in the world by population with 16,753,265 inhabitants in the Territory at the 2011 Census...
, Canada
Canada
Canada is a North American country consisting of ten provinces and three territories. Located in the northern part of the continent, it extends from the Atlantic Ocean in the east to the Pacific Ocean in the west, and northward into the Arctic Ocean...
, Haifa
Haifa
Haifa is the largest city in northern Israel, and the third-largest city in the country, with a population of over 268,000. Another 300,000 people live in towns directly adjacent to the city including the cities of the Krayot, as well as, Tirat Carmel, Daliyat al-Karmel and Nesher...
and Bielefeld
Bielefeld
Bielefeld is an independent city in the Ostwestfalen-Lippe Region in the north-east of North Rhine-Westphalia, Germany. With a population of 323,000, it is also the most populous city in the Regierungsbezirk Detmold...
. He was awarded the Lester R Ford Award from the Mathematical Association of America
Mathematical Association of America
The Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists;...
in 1972 and the Senior Berwick Prize
Berwick Prizes
The Berwick Prize and Senior Berwick Prize are two prizes of the London Mathematical Society awarded in alternating years in memory of William Edward Hodgson Berwick, a previous Vice-President of the LMS. Berwick left some money to be given to the society to establish two prizes...
of the London Mathematical Society
London Mathematical Society
-See also:* American Mathematical Society* Edinburgh Mathematical Society* European Mathematical Society* List of Mathematical Societies* Council for the Mathematical Sciences* BCS-FACS Specialist Group-External links:* * *...
in 1974.
In the early 1980s, funding cuts caused the closure of the small colleges of the University of London. Cohn moved to University College in 1984, together with the two other experts at Bedford on ring theory, Bill Stephenson and Warren Dicks. He became Astor Professor of Mathematics there in 1986. He continued to be a visiting professor, for example to the University of Alberta
University of Alberta
The University of Alberta is a public research university located in Edmonton, Alberta, Canada. Founded in 1908 by Alexander Cameron Rutherford, the first premier of Alberta and Henry Marshall Tory, its first president, it is widely recognized as one of the best universities in Canada...
in 1986 and to Bar Ilan University in 1987. He retired in 1989, but remained active as Professor Emeritus and Honorary Research Fellow until his death.
He was President of the London Mathematical Society, 1982-4, having been its secretary, 1965–67 and a Council member in 1968-71, 1972–75 and 1979-82. He was editor of the Society's Monographs in 1968-77 and 1980-93. He was elected a Fellow of the Royal Society
Royal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...
in 1980 and was on its council, 1985-87. He was a member of the Mathematical Committee of the Science Research Council, 1977-1980. He chaired the National Committee for Mathematics, 1988-9.
Mathematical work
In all, Cohn wrote nearly 200 mathematical papers. He worked in many areas of algebra, mainly in non-commutative ring theory. His first papers, covering many topics, were published in 1952. He generalised a theorem due to Wilhelm MagnusWilhelm Magnus
Wilhelm Magnus was a mathematician. He made important contributions in combinatorial group theory, Lie algebras, mathematical physics, elliptic functions, and the study of tessellations....
, and worked on the structure of tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
spaces. In 1953 he published a joint paper with Kurt Mahler
Kurt Mahler
Kurt Mahler was a mathematician and Fellow of the Royal Society.He was a student at the universities in Frankfurt and Göttingen, graduating with a Ph.D...
on pseudo-valuations and in 1954 he published a work on Lie algebras.
Papers over the next few years covered areas such as group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, field theory
Field theory (mathematics)
Field theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined....
, Lie ring
Lie ring
In mathematics a Lie ring is a structure related to Lie algebras that can arise as a generalisation of Lie algebras, or through the study of the lower central series of groups.- Formal definition :...
s, semigroup
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
s, Abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
s and ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
. He published his first book, on Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s, in 1957. After that, he moved into the areas of Jordan algebra
Jordan algebra
In abstract algebra, a Jordan algebra is an algebra over a field whose multiplication satisfies the following axioms:# xy = yx # = x ....
s, Lie division rings, skew fields, free ideal ring
Free ideal ring
In mathematics, especially in the field of ring theory, a free ideal ring, or fir, is a ring in which all left ideals are free of unique rank. A ring such that all left ideals with at most n generators is free of unique rank is called an n-fir. A semifir is a ring in which all finitely generated...
s and non-commutative unique factorisation domains. He published his second book, Linear equation
Linear equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....
s, in 1958 and his third, Solid geometry
Solid geometry
In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space — for practical purposes the kind of space we live in. It was developed following the development of plane geometry...
, in 1961. Universal algebra appeared in 1965 (second edition 1981). After that, he concentrated on non-commutative ring theory and the theory of algebras.
His monograph Free rings and their relations appeared in 1971. It covered the work of Cohn and others on free associative algebras and related classes of rings, especially free ideal rings. He included all of his own published results on the embedding of rings into skew fields. The second, enlarged edition appeared in 1985.
Cohn also wrote undergraduate textbooks. Algebra volume I appeared in 1974 and volume II in 1977. The second edition, in three volumes, appeared in 1982–1990.
Private life
His recreation was etymology and language in all its forms, and he translated mathematical papers into Spanish, Italian, Russian and Chinese. He married Deirdre Finkel in 1958, and they had two daughters.Publications
- Lie Groups (1957)
- Universal Algebra (1965, 1981)
- Free Rings and Their Relations (1971, 1985)
- Algebra I (1974, 1982)
- Algebra II (1977, 1989)
- Algebra III (1990)
- Algebraic Numbers and Algebraic Functions (1991)
- Elements of Linear Algebra (1994)
- Skew Fields, Theory of General Division Rings (in Encyclopedia of Mathematics and its Applications, vol 57, 1995)
- Introduction to Ring Theory (2000)
- Classic Algebra (2000)
- Basic Algebra (2002)
- Further Algebra and Applications (2003)
- Oxford Dictionary of National BiographyDictionary of National BiographyThe Dictionary of National Biography is a standard work of reference on notable figures from British history, published from 1885...
(contributor, 2004) - Free Ideal Rings and Localization in General Rings (2006)