Tight Closure of Ideals Relative to Modules

Abstract

Abstract. In this paper the dual notion of tight closure of ideals relative to modules is introduced and some related results are obtained.

The Tight Integral Closure of a Set of Ideals Relative to Modules

“…Let M be an R−module. An element x of R is said to be tight dependent on I relative to M if there exists an element c ∈ R • such that (0 : M I[q] ) ⊆ (0 : M cx q ) f or all q 0.It has been proved thatI * [M ] = {x ∈ R : x is tight dependent on I relative to M }is an ideal of R (see[2]). The ideal I* [M ] is called the tight closure of I relative to M .…”