A nil approach to symmetricity of rings

Abstract

We introduce a weakly symmetric ring which is a generalization of a symmetric ring and a strengthening of both a GWS ring and a weakly reversible ring, and investigate properties of the class of this kind of rings. A ring R is called weakly symmetric if for any a, b, c 2 R, abc being nilpotent implies that Racrb is a nil left ideal of R for each r 2 R. Examples are given to show that weakly symmetric rings need to be neither semicommutative nor symmetric. It is proved that the class of weakly symmetric rings lies also between those of 2-primal rings and directly finite rings. We show that for a nil ideal I of a ring R, R is weakly symmetric if and only if R=I is weakly symmetric. If R[x] is weakly symmetric, then R is weakly symmetric, and R[x] is weakly symmetric if and only if R[x; x-1] is weakly symmetric. We prove that a weakly symmetric ring which satises Köthe's conjecture is exactly an NI ring. We also deal with some extensions of weakly symmetric rings such as a Nagata extension, a Dorroh extension.