Vol. 38, No. 1, 1971

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A generalization of commutative and associative rings

Erwin Kleinfeld

Vol. 38 (1971), No. 1, 95–101
Abstract

Let R be a ring satisfying the following three defining relations: (i) (x,y2,x) = y (x,y,x), (ii) (x,y,z) + (y,z,x) + (z,x,y) = 0, and (iii) ((x,y),x,x) = 0, where (a,b,c) = (αb)c a(bc), (a,b) = αb ba, and a b = ab + ba. All three identities follow from commutativity, hence are true in Jordan rings. Besides (i) holds in Lie and alternative rings, (ii) holds in Lie and quasiassociative rings and in alternative rings of characteristic three, while (iii) holds in right alternative rings. The main result is that if R has characteristic 2,3 (that means no elements in R have additive order two or three) and no divisors of zero then R must be either associative or commutative.

Mathematical Subject Classification
Primary: 17E05
Milestones
Received: 2 September 1970
Published: 1 July 1971
Authors
Erwin Kleinfeld