Alternative rings whose symmetric elements are nilpotent or a right multiple is a symmetric idempotent

Abstract

Osborn characterizes those associative rings with involu- tions in which each symmetric element is nilpotent or invertible. Analogous results are obtained for alternative rings. The restriction is further relaxed to require only that each symmetric element is nilpotent or some multiple is a symmetric idempotent. Introduction* J. M. Osborn (10) (11) proved a series of theo- rems concerning the structure of associative rings with involution such that any symmetric element is either nilpotent or invertible. Many generalizations of his results have appeared in the literature for associative rings (a good single reference is Herstein (4)). We begin with a discussion of involutions in the Cayley-Dickson algebras. Then Osborn's results are generalized to alternative rings. Our final result shows that if R is an alternative ring with involution * such that (a) each symmetric element s is either nilpotent or some right multiple of s is a symmetric idempotent and (b) each set of pairwise orthogonal symmetric idempotent has n or less elements, then the quotient ring i?/RadJ? has d.c.c on right ideals. Since a radical free alternative ring with d.c.c. on right ideals is the direct sum of Cayley-Dickso n algebras and simple artinian associative rings, we have a nice description of these quotient rings.