On characterizations of Prüfer v-multiplication domains
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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 Prufer v-multiplication domain if (II-1)w = D for all nonzero finitely generated ideals I of D. In this paper, we show that D is a Prufer 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.
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 Prufer v-multiplication domain if (II-1)w = D for all nonzero finitely generated ideals I of D. In this paper, we show that D is a Prufer 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.
참고문헌 (9)
- D.D. Anderson, "Two star operations and their induced lattices", Comm. Algebra, 28, pp.1475~2461 (2000)
- Nv, "J. Pure Appl. Algebra", 2005, 293~304
- G.W. Chang, "Prufer *-multiplication domains, Nagata rings, and Kronecker function rings", J. Algebra, 319, pp.309~319 (2008)
- D.E. Dobbs, "t-linked overrings and Prufer v-multiplication domains", Comm. Algebra, 17, pp.2835~2852 (1989)
- R. Gilmer, "Multiplicative Ideal Theory" (1972), Dekker
- M. Griffin, "Some results on v-multiplication rings", Canad. J. Math., 19, pp.710~722 (1967)
- C.U. Jensen, "On characterizations of Prufer rings", Math. Scand., 13, pp.90~98 (1963)
- Nv, "J. Algebra", 1989, 151~170
- M. Zafrullah, "Putting t-invertibility to use, in : Non-Noetherian Commutative Ring Theory", Math. Appl., 520, pp.429~457 (2000), Kluwer Acad. Publ.