Equivalent nilpotencies in certain generalized right alternative rings

Harry F. Smith, Jr.

Vol. 98 (1982), No. 2, 459–467

DOI: 10.2140/pjm.1982.98.459

Abstract

A nonassociative ring is called generalized right alternative if it satisfies the identity (wx,y,z) + (w,x,[y,z]) = w(x,y,z) + (w,y,z)x. Generalized right alternative rings which also satisfy ([w,x],y,z) + (w,x,yz) = y(w,x,z) + (w,x,y)z or (x,y,z) + (y,z,x) + (z,x,y) = 0 are known as generalized alternative or generalized (−1,1) rings, respectively. For both these varieties it is proved that either left or right nilpotence implies nilpotence. However, characteristic ≠2 is required for generalized (−1,1) rings in the case of right nilpotence.

Mathematical Subject Classification 2000

Primary: 17A30

Secondary: 17D05

Milestones

Received: 27 October 1980

Published: 1 February 1982