Commutativity theorems for nonassociative rings with a finite division ring homomorphic image

Eugene Carlyle Johnsen, David Lewis Outcalt and Adil Mohamed Yaqub

Vol. 27 (1968), No. 2, 325–332

DOI: 10.2140/pjm.1968.27.325

Abstract

Wedderburn’s Theorem, asserting that a finite associative division ring is necessarily commutative, has been extended to

Theorem 1. Let R be a noncommutative Jordan ring of characteristic not 2, and let I be an ideal in R such that R∕I is a finite division ring of characteristic p > 5 with exactly q elements. Suppose that (i) I is commutative and every associator contained in the ideal generated by I² vanishes, and (ii) x ≡ y (mod I) implies x^q = y^q or both x and y commute with all elements of I. Then R is commutative.

The object of this paper is to extend Theorem 1 in two directions. First we replace the assumption that R is a noncommutative Jordan ring by the weaker assumption that R is power-associative. Next we assume that R is a flexible power-associative ring but replace the hypothesis that every associator in the ideal generated by I² vanishes with the weaker assumption that I is associative. In each case we drop the assumption that R is of characteristic not 2.

Mathematical Subject Classification

Primary: 17.20

Milestones

Received: 3 October 1967

Revised: 5 April 1968

Published: 1 November 1968