On rings with associated elements

Abstract

A principal right ideal of a ring is called uniquely generated if any two elements of the ring that generate the same principal right ideal must be right associated (i.e., if for all a, b in a ring R, aR = bR implies a = bu for some unit u of R). In the present paper, we study uniquely generated modules as a module theoretic version of uniquely generated ideals, and we obtain a characterization of a unit-regular endomorphism ring of a module in terms of certain uniquely generated submodules of the module among some other results: End(M) is unit-regular if and only if End(M) is regular and all M-cyclic submodules of a right R-module M are uniquely generated. We also consider the questions of when an arbitrary element of a ring is associated to an element with a certain property. For example, we consider this question for the ring R[x; sigma]/(x(n+1)), where R is a strongly regular ring with an endomorphism sigma be an endomorphism of R.