DOI: https://doi.org/10.11568/kjm.2010.18.4.

### ON CHARACTERIZATIONS OF PRUFER v-MULTIPLICATION DOMAINS

#### Abstract

Let D be an integral domain with quotient field K,

I(D) be the set of nonzero ideals of D, and w be the star-operation

on D defined by Iw = {x ∈ K|xJ ⊆ I for some J ∈ I(D) such

that J is finitely generated and J −1 = D}. The D is called a Pr ̈

ufer

v-multiplication domain if (II −1 )w = D for all nonzero finitely gen-

erated ideals I of D. In this paper, we show that D is a Pr ̈

ufer

v-multiplication domain if and only if (A ∩ (B + C))w = ((A ∩

B) + (A ∩ C))w for all A, B, C ∈ I(D), if and only if (A(B ∩ C))w =

(AB∩AC)w for all A, B, C ∈ I(D), if and only if ((A+B)(A∩B))w =

(AB)w for all A, B ∈ I(D), if and only if ((A + B) : C)w = ((A :

C) + (B : C))w for all A, B, C ∈ I(D) with C finitely generated, if

and only if ((a : b) + (b : a))w = D for all nonzero a, b ∈ D, if and

only if (A : (B ∩ C))w = ((A : B) + (A : C))w for all A, B, C ∈ I(D)

with B, C finitely generated.

#### Subject classification

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ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr