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Weak injectivity and module classes. (English) Zbl 0934.16004
The concept of weakly injective module was introduced by Jain and López-Permouth and has been widely studied. Let \(E(N)\) denote the injective hull of \(N\). Set \(E_M(N)=\) Trace of \(M\) in \(E(N)=\) the \(M\)-injective hull of \(N\). For any module \(M\), let \(\sigma[M]\) be the full subcategory of \(\text{Mod-}R\), subgenerated by \(M\). A module \(N\in\sigma[M]\) is weakly injective in \(\sigma[M]\), if for every finitely generated submodule \(Y\) of \(E_M(N)\) there exists \(X\subseteq E_M[N]\), such that \(Y\subseteq X\cong N\). A subclass \(\kappa\subseteq\sigma[M]\) is called an \(M\)-natural class if \(\kappa\) is closed under submodules, direct sums, \(M\)-injective hulls and isomorphic copies. The author studies relative weak injectivity in the setting of an \(M\)-natural class. Let \(\kappa\) be an \(M\)-natural class. Characterizations are provided for every module in \(\kappa\) to be weakly injective in \(\sigma[M]\); and also for every direct sum of \(M\)-injective modules in \(\kappa\) to be weakly injective in \(\sigma[M]\). The results restricted to a special module (particular module class \(\kappa\)), yield several known (and new) results as corollaries. These include (known) characterizations of quotient finite dimensional rings, locally quotient finite dimensional modules, semiprime Goldie rings, etc.
MSC:
16D50 Injective modules, self-injective associative rings
16D90 Module categories in associative algebras
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