Some notes on JTTC rings

Abstract

A ring R is called JTTC if for any a∈N(R) and b∈R, (ab)2=ab2a, which is a proper generalization of CN rings. In this paper, we show that (1) a ring R is commutative if and only if (xy)2=xy2x for each x∈SN(R) and y∈SZr(R); (2) R is a JTTC ring if and only if xyx=x2y for each x∈N(R) and y∈SZr(R); (3) R is a reduced ring if and only if T3(R) is a JTTC ring; (4) R is a CN ring if and only if V2(R) is a JTTC ring; (5) R is a commutative reduced ring if and only if TV4(R) is a JTTC ring; (6) R is a commutative ring if and only if G3(R) is a JTTC ring; (7) If R is a JTTC ring containing a von Neumann regular maximal left ideal, then R is a strongly regular ring.