On $w$-Linked Overrings

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.