# Irving Kaplansky

### Quick Info

Toronto, Ontario, Canada

Sherman Oaks, Los Angeles, California, USA

**Kaplansky**has made major contributions to ring theory, group theory and field theory.

### Biography

**Irving Kaplansky**'s parents were Polish and he was born shortly after they had emigrated to Canada. Irving's early interest was music, an interest which he has kept all his life. Anyone who has heard him play the piano at a conference (as I [EFR] have been fortunate enough to do) will have seen that he exudes the same infectious joy of music as he does for mathematics. However Irving knew from a very young age that mathematics, and not music, was to be his life.

He attended the University of Toronto graduating with a B.A. in 1938. He showed his great potential for mathematics at this stage, being on the winning team of the first William Lowell Putnam competition. This is a mathematical contest for students from the USA and Canada.

In 1940 Kaplansky received his M.A. from Toronto, continuing his studies at Harvard University after being awarded a Putnam Fellowship (in fact he was the first recipient of this award). He was awarded a doctorate by Harvard in 1941, the year after he had become a citizen of the United States. His thesis supervisor at Harvard was Mac Lane and Kaplansky's thesis was entitled

*Maximal Fields with Valuations*. Kaplansky was appointed a Benjamin Peirce Instructor in Harvard that year and he continued to hold that post there until 1944.

The year 1944-45 Kaplansky spent in the Applied Mathematics Group of the National Defense Council at Columbia University before moving, in 1945, to the University of Chicago. This was to be the main university where he spent most of his career and where he was promoted to professor. During the years 1962-67 Kaplansky was chairman of the department in Chicago. In 1969 he was appointed George Herbert Mead Distinguished Service Professor at Chicago where he remained till his retirement in 1984. Despite holding important positions he remained accessible to colleagues and students alike, and [1]:-

... one could always rely on his availability and on a challenging idea or question as a result of each conversation.After he retired in 1984, Kaplansky went to California where he became director of the Mathematical Sciences Research Institute at the University of California, Berkeley.

Kaplansky's work in mathematics is wide ranging although mostly it is in areas of algebra. He has made major contributions to ring theory, group theory and field theory. His book

*Infinite Abelian Groups*was written at a time when this area was causing little interest but it has now blossomed into a major area in its own right.

Similarly his many other books are beautiful introductions to various areas of algebra and have been enjoyed for their clarity, style and beauty by large numbers of undergraduate and graduate students. They include

*Fields and rings*(2nd ed, 1972),

*An introduction to differential algebra*(1957),

*Commutative rings*(1970) and

*Lie algebras and locally compact groups*(1971). Kaplansky's books [1]:-

... at a range of levels, are numerous ... [but] they are certainly not ponderous. He is a man of a few words, writing with polished economy to get the important ideas across.Kaplansky has received numerous awards. He has served for many years on the American Mathematical Society, being on the Council in 1951-53, vice-president in 1975, and he was elected president of the Society shortly after he retired during session 1985-86. There are many other ways in which Kaplansky has served the Society, particularly with respect to the American Mathematical Society publications. From 1945 to 1947 and again from 1979 to 1985 he was on the editorial board of the

*Bulletin of the American Mathematical Society*; from 1947 to 1952 he was on the editorial board of the

*Transactions of the American Mathematical Society*; and from 1957 to 1959 he was on the editorial board of the

*Proceedings of the American Mathematical Society*.

Despite this remarkable record of service to the Society, there were still further ways in which Kaplansky used his many talents to its benefit. He served on the Committee on Translations from Russian and other Slavic Languages from 1949 until 1958 and was on the Nominating Committee in 1977-78.

Kaplansky was awarded a Guggenheim Fellowship and elected to the National Academy of Sciences and the American Academy of Arts and Sciences. In 1987 he was made an honorary member of the London Mathematical Society. Two years later, in 1989, the American Mathematical Society awarded Kaplansky their Steele Prize. There are three Steele Prizes awarded for different achievements. Kaplansky was awarded one [1]:-

... in recognition of cumulative influence extending over a career, including the education of doctoral students.The citation for the prize gives an excellent summary of Kaplansky's many achievements. The citation is available from a number of sources, see for example [1]:-

By his energetic example, his enthusiastic exposition and his overall generosity, he has made striking changes in mathematics and has inspired generations of younger mathematicians. His early works range over number theory, statistics, combinatorics, game theory, as well as his principal interest of commutative algebra. He completed the solution of Kurosh's problem on algebraic algebras of bounded degree, where Jacobson had made a decisive reduction, and considered numerous questions in the area of Banach algebras, always from the algebraist's viewpoint. ...Kaplansky could not be present at the Summer Meeting of the American Mathematical Society in 1989 to reply in person to this citation. However, he did give a written response which was read at the meeting. In this response he showed his modestly by claiming that the "citation ... is too flattering" but he also gave some good advice which he wanted to put into print and it is well worth repeating here [1]:-

As commutative algebra took on new life with the infusion of homological methods, he turned his interest once more in this direction, always trying to see past the formalism into "what was really going on". His remarkable success in doing so is witnessed by his publications from the later fifties onwards and the influence they have had on other writers. ...

... spend some time every day learning something new that is disjoint from the problem on which you are currently working (remember that the disjointness may be temporary), and read the masters.

### References (show)

- 1989 Steele prizes awarded at Summer Meeting in Boulder,
*Notices Amer. Math. Soc.***36**(7) (1989), 831- 836.

### Additional Resources (show)

Other pages about Irving Kaplansky:

Other websites about Irving Kaplansky:

### Honours (show)

Honours awarded to Irving Kaplansky

### Cross-references (show)

Written by J J O'Connor and E F Robertson

Last Update May 2000

Last Update May 2000