Δ-Reductions and Δ-Closures of Ideals with Respect to an Artinian Module

Abstract

Let A denote an Artinian module over a commutative ring R and let be a multiplicatively closed set of nonzero ideals of R. The purpose of this article is to show that the Artinian property of A allows one to develop satisfactory important concepts of -reduction and -closure of an ideal w.r.t. A whose properties reflect some of those of the usual concepts of -reduction and -closure of an ideal in a commutative ring introduced by L. J. Ratliff. Among other things, it is shown that the operation I −→ I A is a semiprime operation on the set of ideals I of R that satisfies a partial cancelation law. Also, if any ideal in is finitely generated, then I A is decomposable and the associated primes of I A are also associated primes of I A K and IK A for all K ∈ . Finally, in the case when R is complete local (Noetherian) and every ideal in is A-coregular, then we show that the sequence Ass R R/ I n A n≥1 of associated prime ideals is increasing and eventually constant.