The socle of an alternative ring

Abstract

The main facts about the minimal ideals and minimal right ideals of an associative ring are well known. In this paper we prove corresponding results for an alternative ring R. We make no restriction on the characteristic of R, but will often impose restrictions of semiprimeness type. (R is semiprime provided it has no ideal T such that T # (0) = P). Throughout this paper “ring” will mean “alternative ring”, and R will be a ring. We write A < R (A <, R; A <, R) to denote that A is an ideal (right ideal, left ideal) of R. If A is a minimal element of the set {M : (0) # M < R} (respectively, {M : (0) # M <,. R}), partially ordered by inclusion, we say that A is a minimal ideal (respectively, minimal fight ideal) of R, and write A <, R (respectively, A <,, R). Similarly for A <,, R. In Section 2, we show that if A <, R then either A2 = (0) or A is simple; this result is due in part to Zhevlakov. In Section 3 we similarly characterize A if A &. R. Roughly, if A2 # (0), then A is a minimal right ideal of the ideal C it generates in R, and, in general, C is simple. We also show that A is of the form eR, where e is a nuclear idempotent. In Section 4 we consider the right socle of R : S,(R) = Z(M : M <,, R). Under a suitable weak condition S,(R) is a two-sided ideal, and under a slightly stronger condition (weaker than semiprimeness of R) it coincides with the analogously defined left socle S,(R). In this situation we define the socle of R, S(R), to be S,(R) = S,(R). A structure theorem is proved for S,(R) which (like the results for minimal ideals and minimal right ideals) is particularly informative if R is purely alternative (free of nuclear ideals): that is, in some sense at the opposite extreme from being associative. We also show that S,(R) annihilates the Smiley radical M(R), thus justifying Baer’s name “antiradical” for S,(R). One result of Section 4 is that if A < R and R is semiprime, then S(A) = A n S(R). I n ec ion 5 we consider when a corresponding result S t