Strong Mori modules over an integral domain
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Let D be an integral domain with quotient field K, M a torsion-free D-module, X an indeterminate, and Nv = {f∈D[X] | c(f)v = D}. Let q(M) = M D K and MwD = {x ∈ q(M) | xJ ⊆ M for a nonzero finitely generated ideal J of D with Jv = D}. In this paper, we show that MwD = M[X]Nv ∩ q(M) and (M[X])w D[X] ∩ q(M)[X] = MwD[X] = M[X]Nv ∩ q(M)[X]. Using these results, we prove that M is a strong Mori D-module if and only if M[X] is a strong Mori D[X]- module if and only if M[X]Nv is a Noetherian D[X]Nv -module. This is a generalization of the fact that D is a strong Mori domain if and only if D[X] is a strong Mori domain if and only if D[X]Nv is a Noetherian domain.
Abstract
Let D be an integral domain with quotient field K, M a torsion-free D-module, X an indeterminate, and Nv = {f∈D[X] | c(f)v = D}. Let q(M) = M D K and MwD = {x ∈ q(M) | xJ ⊆ M for a nonzero finitely generated ideal J of D with Jv = D}. In this paper, we show that MwD = M[X]Nv ∩ q(M) and (M[X])w D[X] ∩ q(M)[X] = MwD[X] = M[X]Nv ∩ q(M)[X]. Using these results, we prove that M is a strong Mori D-module if and only if M[X] is a strong Mori D[X]- module if and only if M[X]Nv is a Noetherian D[X]Nv -module. This is a generalization of the fact that D is a strong Mori domain if and only if D[X] is a strong Mori domain if and only if D[X]Nv is a Noetherian domain.
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