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Keywords:
$w$-universal injective; $w$-flat module; $w$-IF ring; $w$-coherent ring
$w$-universal injective; $w$-flat module; $w$-IF ring; $w$-coherent ring
Summary:
Let $R$ be a commutative ring and $w$ be the $w$-operation on $R$. We introduce the concept of $w$-universal injective modules and establish their fundamental properties. It is shown that the product of $E(R/{\frak m})$, where ${\frak m}$ ranges over maximal $w$-ideals of $R$, is a \hbox {$w$-universal} injective $w$-module over $R$, albeit not necessarily a universal injective \hbox {$R$-module}. As applications, we characterize $w$-IF rings and $w$-coherent rings using $w$-universal injective modules. Specifically, we demonstrate that $R$ is a $w$-IF ring if and only if $R$ is $w$-coherent and $E(R/{\frak m})$ is a flat $R$-module for every ${\frak m} \in w \text {-Max}(R)$. These results extend existing results and provide deeper insights into the structure of $w$-modules.
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