Strong Mori modules over an integral domain
Bull. Korean Math. Soc. 2013 Vol. 50, No. 6, 1905-1914
https://doi.org/10.4134/BKMS.2013.50.6.1905
Published online November 1, 2013
Gyu Whan Chang
Incheon National University
Abstract : Let $D$ be an integral domain with quotient field $K$, $M$ a torsion-free $D$-module, $X$ an indeterminate, and $N_v = \{f \in D[X]~|~ c(f)_v $ $= D\}$. Let $q(M)= M \otimes_D K$ and $M_{w_D} = \{x \in q(M)~|~ xJ \subseteq M$ for a nonzero finitely generated ideal $J$ of $D$ with $J_v = D\}$. In this paper, we show that $M_{w_D} = M[X]_{N_v} \cap q(M)$ and $(M[X])_{w_{D[X]}} \cap q(M)[X] = M_{w_D}[X] = M[X]_{N_v} \cap q(M)[X]$. 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 $M[X]_{N_v}$ is a Noetherian $D[X]_{N_v}$-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 $D[X]_{N_v}$ is a Noetherian domain.
Keywords : polynomial module, Noetherian module, strong Mori module
MSC numbers : 13A15
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