ON ⋆-COLON-MULTIPLICATION DOMAINS

Let ⋆ be a star operation on an integral domain R. An ideal A is a ⋆-colon-multiplication ideal if A⋆ = (B(A : B))⋆ for all fractional ideal B of R. We prove that every maximal ideal of R is a ⋆-colon-multiplication ideal if and only if R is a ⋆-CICD or R is a local ⋆-MTP domain. It is also shown that every ideal of R is ⋆-colon-multiplication if and only if R is a ⋆-CICD.

Note on Colon-Multiplication Domains

LetRbe an integral domain with quotient fieldL.Call a nonzero (fractional) idealAofRa colon-multiplication ideal any idealA, such thatB(A:B)=Afor every nonzero (fractional) idealBofR.In this note, we characterize integral domains for which every maximal ideal (resp., every nonzero ideal) is a colon-multiplication ideal. It turns that this notion unifies Dedekind andMTPdomains.

Multiplication Ideals, Multiplication Rings, and the Ring R(X)

Let R be a commutative ring with an identity. An ideal A of R is called a multiplication ideal if for every ideal B ⊆ A there exists an ideal C such that B = AC. A ring R is called a multiplication ring if all its ideals are multiplication ideals. A ring R is called an almost multiplication ring if RM is a multiplication ring for every maximal ideal M of R. Multiplication rings and almost multiplication rings have been extensively studied—for example, see [4; 8; 9; 11; 12; 15; and 16].