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On weakly projective and weakly injective modules. (English) Zbl 1101.16004
Let $$M$$ be a right module over an associative ring $$R$$ with identity. The class of all submodules of $$M$$-generated modules is denoted by $$\sigma[M]$$. For a module $$X$$ the injective hull of $$X$$ is denoted by $$E(X)$$ and the $$M$$-injective hull of $$X$$ is denoted by $$\widehat X=\{f(M),\;f\in\operatorname{Hom}(M,E(X))\}$$.
Recall, that a module $$Q$$ is called weakly $$N$$-injective in $$\sigma[M]$$, $$N\in\sigma[M]$$, if for every homomorphism $$\varphi\colon N\to\widehat Q$$ there exists a homomorphism $$\widehat\varphi\colon N\to Q$$ and a monomorphism $$\sigma\colon Q\to\widehat Q$$ such that $$\varphi=\sigma\widehat\varphi$$. A module $$Q\in\sigma[M]$$ is called weakly injective in $$\sigma[M]$$ if for every finitely generated submodule $$N$$ of the $$M$$-injective hull $$\widehat Q$$, $$N$$ is contained in a submodule $$Y$$ of $$\widehat Q$$ with $$Y\cong Q$$.
Similarly, $$Q$$ is called weakly $$N$$-projective in $$\sigma[M]$$, $$N\in\sigma[M]$$, if $$Q$$ has a $$\sigma[M]$$-projective cover $$P(Q)$$ and for every homomorphism $$\varphi\colon P(Q)\to N$$ there exists a homomorphism $$\widehat\varphi\colon Q\to N$$ and an epimorphism $$\sigma\colon P(Q)\to Q$$ with $$\varphi=\widehat\varphi\sigma$$. A module $$Q\in\sigma[M]$$ is called weakly projective in $$\sigma[M]$$ if it is weakly $$N$$-projective for all finitely $$M$$-generated modules $$N$$ in $$\sigma[M]$$. A module $$M$$ is called locally qfd if every finitely generated module $$N\in\sigma[M]$$ has finite uniform dimension.
Among other results it is shown that if $$M$$ is a locally qfd module then there exists a module $$K\in\sigma[M]$$ such that $$Q=K\oplus N$$ is a weakly injective module for every module $$N\in\sigma[M]$$ (Theorem 2.7). If $$M$$ is projective and right perfect in $$\sigma[M]$$ then there exists a module $$K\in\sigma[M]$$ such that $$K\oplus X$$ is a weakly projective module in $$\sigma[M]$$ for every module $$X\in\sigma[M]$$ (Theorem 2.9). – In the final part of the paper for some classes $$\mathcal M$$ of modules in $$\sigma[M]$$ the direct sums of modules from $$\mathcal M$$ satisfying the module property $$\mathbb{P}$$ in $$\sigma[M]$$ are investigated. Especially, the classes of locally countably thick modules and weakly semisimple modules are characterized.
##### MSC:
 16D50 Injective modules, self-injective associative rings 16D90 Module categories in associative algebras 16D40 Free, projective, and flat modules and ideals in associative algebras 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
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