multiplication ideal
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2012 ◽  
Vol 12 (02) ◽  
pp. 1250156
Author(s):  
OLIVIER A. HEUBO-KWEGNA

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.


Author(s):  
A. Mimouni

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.


1987 ◽  
Vol 48 (4) ◽  
pp. 298-302 ◽  
Author(s):  
Scott M. Farrand ◽  
David R. Finston

1976 ◽  
Vol 28 (4) ◽  
pp. 760-768 ◽  
Author(s):  
D. D. Anderson

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].


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