On a property of polynomial rings over reversible rings

Abstract

In this article, we first observe a sort of property of ideals generated by zero-dividing polynomials over reversible rings, in relation with products of ideals at zero. As a natural consequence of this result, we introduce the concept of a partially reflexive ring that is related to the reflexive ring property. It is shown that abelian pi-regular rings are partially reflexive when Jacobson radicals are nilpotent. A ring R, in which the Jacobson radical J(R) is nilpotent and is simple, is shown to be partially reflexive; and it is proved that the polynomial ring over R is also partially reflexive. The structures of several kinds of algebraic systems are investigated with respect to the partial reflexivity.