Regular elements in rings with involution

Abstract

The purpose of this paper is to determine when a symmetric element, regular with respect to other symmetrics, is regular in the ring. This result is true for simple rings, for prime rings with either Goldie chain condition, and for semiprime Goldie rings. Examples are given to show that these results are the best that can be hoped for. The structure of a ring with involution satisfying the property that no two nonzero symmetric elements annihilate one another was determined for 2-torsion. free rings by Lanski in [4], and in general, by Herstein in [2]. It happens, at least for semiprime rings which are 2-torsion-free, that this assumption on the symmetric elements implies that each nonzero symmetric element is regular in the whole ring. It is our purpose here to determine conditions sufficient to imply that any single symmetric element which annihilates no nonzero symmetric element is regular in the whole ring. Throughout this work, R will denote a ring with involution * and S = r e RIr*= rj, the set of symmetric elements of R. To avoid confusion, we may write S(A) for the symmetric elements of some ring A, with involution. We shall say that an element y is regular in S to mean that yt = 0 implies t = 0, for t E S. We are interested in determining when an element y e S, which is regular in S, must also be regular in R. Of course, we cannot hope to draw such a conclusion unless R is semiprime. A counterexample can be found easily by taking R @ N, for N a 2-torsion-free ring with trivial multiplication, and setting x* = -x for x e N. It is also easy to find examples where R has 1, and so N is not a direct summand. Let R = F[x, y. z]/(z2, zx, zy), for F a field with char F , 2, and set z*-z, x y and y*= x. One condition on R which allows us to conclude that elements regular in S are regular in R is given in our first observation. Received by the editors September 7, 1973. AMS(MOS) subject classifications (1970). Primary 16A28; Secondary 16A12, 16A46, 16A40, 16A38. (1) This work was supported by NSF Grant GP 38601. Copyright