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 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.

Key Words:

• Artinian module
• Associated primes
• Δ-closure
• Δ-reduction

2000 Mathematics Subject Classification:

• Primary 13A30, 13E05
• Secondary 13A02

ACKNOWLEDGMENTS

This work has been supported by the Research Institute for Fundamental Sciences, Tabriz, Iran. The authors are deeply grateful to the referee for his or her valuable suggestions on the article and for drawing the author' attention to Remark 4.5 and Theorem 3.8. Part of this article was written while visiting the Martin-Luther-Universität Halle-Wittenberg. The authors would like to express their sincere thank to the above mentioned University for the hospitality and facilities. The authors also would like to thank Professor P. Schenzel for his useful suggestions and the many helpful discussions.

Notes

Communicated by S. Goro.