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.