Localization and Artinian quotient rings

Abstract

For a non-commutative ring there exists not always the classical quotient ring, while there exists always the maximal quotient ring. In the case of a semiprime ring, howevel; to have a semi-simple Artinian maximal quotient ring implies the existence of the classical quotient ring (cf. Johnson [6, 4.4 Theorem] and Sandomierski [17, Theorem 1.6]). Therefore avoiding the use of the Ore condition it may be possible to find some conditions, under which the existence of the classical quotient ring is assured by requiring the chain condition on a ring which is constructed by a suitable localization. The purpose of the present paper is to provide such conditions and localizations. For a ring with an identity every idempotent topologizing filter ~is defined by a finitely cogenerating injective right R-module W in the following way: ~ = {right ideal D of R I HomR (R/D, W) = 0}, and we have the localizing functor H such that H(X)=~ Horn R (D, X/X~), where X is a right R-module DE~ and X ~ = {xeXIAnnR x e ~ } . H(R) has a ring-structure and H(X) becomes a right H(R)-module in a natural way. Recently, Morita [14] has shown that H(R) and the double centralizer Q of W R are identical and H is exact. With these notations we can state our main theorem in w 3: The following statements I to III are equivalent.