A study on modularization of $w$-linked homomorphisms
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Let $R$ and $T$ be commutative rings, and let $\phi: R\ra T$ be a $w$-linked homomorphism. Let $w_{\phi}$ be a star operation on $T$ with respect to the $w$-linked homomorphism $\phi$. Using the $w_{\phi}$-operation, we introduce concepts such as $\GV_{\phi}$-torsion modules, $\GV_{\phi}$-torsion-free modules, $w_{\phi}$-modules, $\GV_{\phi}$-divisible modules, $w_{\phi}$-finitely generated modules, and $w_{\phi}$-finitely presented modules. Furthermore, we define $w_{\phi}$-flat modules and $w_{\phi}$-coherent rings in a natural manner and study them with the aid of $w_{\phi}$-Nataga rings. In particular, we provide a new version of Chase's theorem for $w_{\phi}$-coherent rings and an exact version of the Eakin-Nagata theorem for $w$-coherent rings.
Abstract
Let $R$ and $T$ be commutative rings, and let $\phi: R\ra T$ be a $w$-linked homomorphism. Let $w_{\phi}$ be a star operation on $T$ with respect to the $w$-linked homomorphism $\phi$. Using the $w_{\phi}$-operation, we introduce concepts such as $\GV_{\phi}$-torsion modules, $\GV_{\phi}$-torsion-free modules, $w_{\phi}$-modules, $\GV_{\phi}$-divisible modules, $w_{\phi}$-finitely generated modules, and $w_{\phi}$-finitely presented modules. Furthermore, we define $w_{\phi}$-flat modules and $w_{\phi}$-coherent rings in a natural manner and study them with the aid of $w_{\phi}$-Nataga rings. In particular, we provide a new version of Chase's theorem for $w_{\phi}$-coherent rings and an exact version of the Eakin-Nagata theorem for $w$-coherent rings.