Geodesic
A note on reductions of ideals relative to an Artinian module
Glasgow mathematical journal, Tome 35 (1993) no. 2, pp. 219-224

Voir la notice de l'article provenant de la source Cambridge University Press

The concept of reduction and integral closure of ideals relative to Artinian modules were introduced in [7]; and we summarize some of the main aspects now.Let A be a commutative ring (with non-zero identity) and let a, b be ideals of A. Suppose that M is an Artinian module over A. We say that a is a reduction of b relative to M if a ⊆ b and there is a positive integer s such that)O:Mabs)=(O:Mbs+l).An element x of A is said to be integrally dependent on a relative to M if there exists n y N(where N denotes the set of positive integers) such thatIt is shown that this is the case if and only if a is a reduction of a+Ax relative to M; moreoverᾱ={x ɛ A: xis integrally dependent on a relative to M}is an ideal of A called the integral closure of a relative to M and is the unique maximal member of℘ = {b: b is an ideal of A which has a as a reduction relative to M}. 
Taherizadeh, A.-J. A note on reductions of ideals relative to an Artinian module. Glasgow mathematical journal, Tome 35 (1993) no. 2, pp. 219-224. doi: 10.1017/S0017089500009770
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