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A note on reductions of ideals relative to an Artinian module

Published online by Cambridge University Press:  18 May 2009

A.-J. Taherizadeh
A.-J. Taherizadeh
Affiliation:
Department of Mathematics, Teacher Training University, 49 Mofateh Ave, Tehran, Iran
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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 that

It is shown that this is the case if and only if a is a reduction of a+Ax relative to M; moreover

ᾱ={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}.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 35 , Issue 2 , May 1993 , pp. 219 - 224
Copyright
Copyright © Glasgow Mathematical Journal Trust 1993

References

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