ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

2. Algebra

1991 MS Classification: 16D70, 16N40 Nagahara, Takasi, Prof. Emer, Okayama Univ, Japan, Yaqob, Adil Univ. of California, Santa Barbara, USA Classification of rings satisfying some generalized Herstein’s condition. Let R be a ring which is not necessarily commutative. Let C and N be the center of R and the set of all nilpotents of R respectively. In [1] Herstein generalized a theorem of Jacobson [2, Theorem 11) His theorem states that if for every a ∈ R, there exists an integer n(a) > 1, depending on a such that a^{n(a)} − a ∈ C, then R is commutative. The following result is proved here: If R ≠ C and for every a ∈ R\(C ∪ N), there exists an integer n(a) > 1, depending on a, such that a^{n(a)} − a ∈ C, then R is one of five types whose structures can be determined completely. References: [1]: L. N. Herstein. A generalization of a theorem of Jacobson, III, Amer. J. Math. 75 (1953), 105 - 111. [2]: N. Jacobson, Structure theory for algebraic algebras of bounded algebras of bounded degree, Ann. of Math. 46, (1945), 695 - 707.