Invariant Subrings in Rings with Involution

Abstract

The purpose of this paper is to consider, in rings with involution, the structure of those subrings which are invariant under Lie commutation with [K, K]. Our goal is to find conditions which force such subrings to contain a noncentral ideal of the ring. Of course, the subring itself may lie in the center. Orders in 4 × 4 matrix rings over fields are known to provide examples of invariant subrings which are not central and contain no ideal (see [1, 40] or [5]). Except for subdirect products of these two kinds of “counter-examples”, we show that in semi-prime rings, invariant subrings do contain noncentral ideals. This generalizes work of Herstein [4] in two directions by considering semi-prime rings rather than simple rings, and by using [K, K] instead of K.

Commutation with skew elements in rings with involution

“…It can be shown that this condition forces J + P to satisfy S 4 , although one can get directly that J + P satisfies S 8 by using Lemma 1,applying [10;Lemma 3], and then applying [7;Lemma 2,p. 735].…”

“…For prime rings, the Lie ideals of both K and [K, JKΓ] were studied by Erickson [3], and an investigation of additive subgroups of K invariant under commutation with [K, K] in semi-prime rings was made in [7]. This was followed by a description of arbitrary additive subgroups invariant under commutation with [K f K] [9], and of subgroups of K invariant under commutation with higher commutators of K [10].…”