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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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