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

Abstract

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