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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.
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참고문헌 (9)
참고문헌 신청
B.G. Kang / 1989 / Prufer v-multiplication domains and the ring R[X]Nv / J. Algebra 123 : 151 ~ 170
C.U. Jensen / 1963 / On characterizations of Prufer rings / Math. Scand. 13 : 90 ~ 98
D.D. Anderson / 2000 / Two star operations and their induced lattices / Comm. Algebra 28 : 1475 ~ 2461
D.E. Dobbs / 1989 / t-linked overrings and Prufer v-multiplication domains / Comm. Algebra 17 : 2835 ~ 2852
G.W. Chang / 2005 / Strong Mori domains and the ring D[X]Nv / J. Pure Appl. Algebra 197 : 293 ~ 304
C.U. Jensen / 1963 / On characterizations of Prufer rings / Math. Scand. 13 : 90 ~ 98
D.D. Anderson / 2000 / Two star operations and their induced lattices / Comm. Algebra 28 : 1475 ~ 2461
D.E. Dobbs / 1989 / t-linked overrings and Prufer v-multiplication domains / Comm. Algebra 17 : 2835 ~ 2852
G.W. Chang / 2005 / Strong Mori domains and the ring D[X]Nv / J. Pure Appl. Algebra 197 : 293 ~ 304
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