| On $w$-Linked Overrings |
| Received:January 12, 2009 Revised:January 18, 2010 |
| Key Words:
GV-ideal w-module w-linked w-Noetherian ring.
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| Fund Project:Supported by the National Natural Science Foundation of China (Grant No.10671137) and by Specialized Research Fund for the Doctoral Program of Higher Education (Grant No.20060636001). |
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| Abstract: |
| Let $R\subseteq T$ be an extension of commutative rings. $T$ is called $w$-linked over $R$ if $T$ as an $R$-module is a $w$-module. In the case of $R\subseteq T\subseteq Q_0(R)$, $T$ is called a $w$-linked overring of $R$. As a generalization of Wang-McCsland-Park-Chang Theorem, we show that if $R$ is a reduced ring, then $R$ is a $w$-Noetherian ring with $w$-$\dim(R)\leqslant 1$ if and only if each $w$-linked overring $T$ of $R$ is a $w$-Noetherian ring with $w$-$\dim(T)\leqslant 1$. In particular, $R$ is a $w$-Noetherian ring with $w$-$\dim(R)=0$ if and only if $R$ is an Artinian ring. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2011.02.018 |
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