Contemporary Mathematics Essential Extensions of a Direct Sum of Simple Modules

Abstract. It is known that every essential extension of a direct sum of injective hulls of simple R-modules is a direct sum of injective R-modules if and only if the ring R is right noetherian. The purpose of this paper is to study the rings R having the property: every essential extension of a direct sum of simple R-modules is a direct sum of quasi-injective R-modules. A commutative ring with this property is known to be an artinian principal ideal ring. In the present paper, we show that such a ring is directly finite. For a right nonsingular ring R with this property, we show that the maximal right ring of quotients Qrmax(R) is the direct product of a finite number of matrix rings over abelian regular self-injective rings. For a von-Neumann regular ring R, we show that R is noetherian if and only if every essential extension of a direct sum of simple R-modules is a quasi-injective R-module. 1