Irving Kaplansky

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Irving Kaplansky
Irving Kaplansky2.jpg
Born March 22, 1917
Died June 25, 2006 (age 89)
Los Angeles
Nationality Canada, American
Fields Mathematics
Institutions University of Chicago
Alma mater Harvard University
Doctoral advisor Saunders Mac Lane
Doctoral students Hyman Bass
H. Arlen Brown
Jacob Feldman
Eben Matlis
Donald Ornstein
Alex F. T. W. Rosenberg
Harold Widom
Fred Wright, Jr.
Known for group theory
ring theory
operator algebras
field theory

Irving Kaplansky (March 22, 1917 – June 25, 2006) was a Canadian mathematician.


Kaplansky (or "Kap," as his friends and colleagues called him) was born in Toronto, Ontario, Canada, to Polish-Jewish immigrants;[1][2] his father worked as a tailor, and his mother ran a grocery and, eventually, a chain of bakeries.[3][4][5] He attended the University of Toronto as an undergraduate. In his senior year, he competed in the first William Lowell Putnam Mathematical Competition, becoming one of the first five recipients of the Putnam Fellowship, which paid for graduate studies at Harvard University.[3] After receiving his Ph.D. from Harvard in 1941[6] as Saunders Mac Lane's first student, he remained at Harvard as a Benjamin Pierce Instructor, and in 1944 moved with Mac Lane to Columbia University for a year.[3] He was professor of mathematics at the University of Chicago from 1945 to 1984, and Chair of the department from 1962 to 1967. Kaplansky's many doctoral students included Hyman Bass, Donald Ornstein, and Harold Widom. Kaplansky was the Director of the Mathematical Sciences Research Institute from 1984 to 1992, and the President of the American Mathematical Society from 1985 to 1986.

Kaplansky was also an accomplished amateur musician. He studied piano until the age of 15, earned money in high school as a dance band musician, taught Tom Lehrer, and played in Harvard's jazz band in graduate school. He also had a regular program on Harvard's student radio station. After moving to the University of Chicago, he stopped playing for two decades, but then returned to music as an accompanist for student-run Gilbert and Sullivan productions and as a calliope player in football game parades.[3] He often composed music based on mathematical themes. One of those compositions, A Song About Pi, is a melody based on assigning notes to the first 14 decimal places of pi, and has occasionally been performed by his daughter, singer-songwriter Lucy Kaplansky.

Mathematical contributions[edit]

Kaplansky made major contributions to group theory, ring theory, the theory of operator algebras and field theory. He published more than 150 papers and worked with at least 20 co-authors.

Awards and honors[edit]

Kaplansky was a member of the National Academy of Sciences and the American Academy of Arts and Sciences.

Selected publications[edit]


  • Kaplansky, Irving (1954). Infinite Abelian groups. [7] revised edn. 1971 with several later reprintings
  • —— (1955). An introduction to differential algebra. University of Chicago Press.  2nd edn. Paris: Hermann. 1957. 
  • —— (1966). Introdução à teoria de Galois, por I. Kaplansky. Pref. de Elon Lages Lima. 
  • —— (1968). Rings of operators. 
  • —— (1969). Fields and rings.  2nd edn. 1972
  • —— (1969). Linear algebra and geometry; a second course.  revised edn. 1974
  • —— (1970). Algebraic and analytic aspects of operator algebras. 
  • —— (1971). Lie Algebras and Locally Compact Groups. University of Chicago Press. ISBN 0-226-42453-7.  several later reprintings
  • —— (1972). Set theory and metric spaces.  2nd edn. 1977
  • —— (September 1974). Commutative Rings. Lectures in Mathematics. University of Chicago Press. ISBN 0-226-42454-5.  1st edn. 1966; revised 1974 with several later reprintings

with I. N. Herstein: —— (1974). Matters mathematical.  2nd edn. 1978


  • "Symbolic solution of certain problems in permutations". Bull. Amer. Math. Soc. 50: 906–914. 1944. MR 0011393. 
  • "A note on groups without isomorphic subgroups". Bull. Amer. Math. Soc. 51: 529–530. 1945. MR 0012267. 
  • with I. S. Cohen: "Rings with a finite number of primes. I". Trans. Amer. Math. Soc. 60: 468–477. 1946. MR 0019595. 
  • "On a problem of Kurosch and Jacobson". Bull. Amer. Math. Soc. 52: 496–500. 1946. MR 0016758. 
  • "Lattices of continuous functions". Bull. Amer. Math. Soc. 53: 617–623. 1947. MR 0020715. 
  • with Richard F. Arens: "Topological representations of algebras". Trans. Amer. Math. Soc. 63: 457–481. 1948. MR 0025453. 
  • "Rings with a polynomial identity". Bull. Amer. Math. Soc. 54: 575–580. 1948. MR 0025451. 
  • "Topological rings". Bull. Amer. Math. Soc. 54: 909–916. 1948. MR 0027269. 
  • "Elementary divisors and modules". Trans. Amer. Math. Soc. 66: 464–491. 1949. MR 0031470. 
  • "Primary ideals in group algebras". Proc Natl Acad Sci U S A 35 (3): 133–136. 1949. PMC 1062983. 
  • "Topological representations of algebras. II". Trans. Amer. Math. Soc. 68: 62–75. 1950. MR 0032612. 
  • "The Weierstrass theorem in fields with valuations". Proc. Amer. Math. Soc. 1: 356–357. 1950. MR 0035760. 
  • "The structure of certain operator algebras". Trans. Amer. Math. Soc. 70: 219–255. 1951. MR 0042066. 
  • "Modules over Dedekind rings and valuations rings". Trans. Amer. Math. Soc. 72: 327–340. 1952. MR 0046349. 
  • "Orthogonal similarity in infinite dimensional spaces". Proc. Amer. Math. Soc. 3: 16–25. 1952. MR 0046564. 
  • "Symmetry of Banach algebras". Proc. Amer. Math. Soc. 3: 396–399. 1952. MR 0048711. 
  • "Some results on abelian groups". Proc Natl Acad Sci U S A 38 (6): 538–540. 1952. PMC 1063607. 
  • "Infinite dimensional quadratic forms admitting composition". Proc. Amer. Math. Soc. 4: 956–960. 1953. MR 0059895. 
  • "Dual modules over a valuation ring. I". Proc. Amer. Math. Soc. 4: 213–219. 1953. MR 0053092. 
  • "Lie algebras of characteristic p". Trans. Amer. Math. Soc. 89: 149–183. 1958. MR 0099359. 
  • "Decomposability of modules". Proc. Amer. Math. Soc. 13: 532–535. 1962. MR 0137738. 
  • "Superalgebras". Pacific J. Math 86 (1): 93–98. 1980. 
  • "A quasi-commutative ring that is not neo-commutative". Proc. Amer. Math. Soc. 122: 321. 1994. MR 1257114. 
  • "The forms x+32y2 and x+64y2 ". Proc. Amer. Math. Soc. 131: 2299–2300. 2003. MR 1963780. 

See also[edit]


  1. ^ Irving Kaplansky Memoir by Nancy E. Albert
  2. ^ Making Family Stories into Art
  3. ^ a b c d Albers, Donald J.; Alexanderson, Gerald L.; Reid, Constance, eds. (1990), "Irving Kaplansky", More Mathematical People, Harcourt Brace Jovanovich, pp. 118–136 .
  4. ^ Irving Kaplansky — mathematician and author
  5. ^ In memoriam: Irving Kaplansky
  6. ^ Irving Kaplansky at the Mathematics Genealogy Project
  7. ^ Baer, Reinhold (1955). "Review: Infinite abelian groups by I. Kaplansky". Bull. Amer. Math. Soc. 61 (1): 88–89. 


External links[edit]