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.
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机译：假设$ D $是商域$ K $，$ M $是无扭转$ D $ -module，$ X $是不确定的且$ N_v = {f in D [X]〜|〜 c（f）_v $ $ = D } $。设$ q（M）= M otimes_D K $和$ M_ {w_D} = {x in q（M）〜|〜xJ subseteq M $对于一个非零有限生成的$ D $理想$ J $ $ J_v = D } $。在本文中，我们证明$ M_ {w_D} = M [X] _ {N_v} cap q（M）$和$（M [X]）_ {w_ {D [X]}} cap q（ M）[X] = M_ {w_D} [X] = M [X] _ {N_v} cap q（M）[X] $。使用这些结果，我们证明，当且仅当$ M [X] $是且仅当$ M [X]时，$ M $是一个强大的Mori $ D [X] $-模块。 _ {N_v} $是Noetherian $ D [X] _ {N_v} $模块。这是对以下事实的概括：$ D $是强Mori域，且仅当$ D [X] $是强Mori域时且仅当$ D [X] _ {N_v} $是Noetherian域时，才是。
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