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A Theorem on Rings

Published online by Cambridge University Press:  20 November 2018

I. N. Herstein
I. N. Herstein*
Affiliation:
Cowles Commision for Research in EconomicsandThe University of Chicago
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In a recent paper, Kaplansky [2] proved the following theorem: Let R be a ring with centre Z, and such that xn(x) ∈ Z for every x∈ R. If R, in addition, is semi-simple then it is also commutative

The existence of non-commutative rings in which every element is nilpotent rules out the possibility of extending this result to all rings. One might hope, however, that if R is such that xn(x) ∈ Z for all x ∈ R and the nilpotent elements of R are reasonably “well-behaved,” then Kaplansky's theorem should be true without the restriction of semi-simplicity.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 5 , 1953 , pp. 238 - 241
Copyright
Copyright © Canadian Mathematical Society 1953

References

1. Baer, Reinhold, Radical ideals, Amer. J. Math., 65 (1943), 540.Google Scholar
2. Kaplansky, Irving, A theorem on division rings, Can. J. Math., 3 (1951), 290–292.Google Scholar
3. Köthe, G., Die Struktur der Ringe, deren Restklassenring nach dem Radikal vollstöndig reduzibel ist, Math. Z., 32 (1930), 170.Google Scholar