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Weak relative injective $$M$$-subgenerated modules. (English) Zbl 0934.16002
Jain, S. K. (ed.) et al., Advances in ring theory. Proceedings of the ring theory section of the 23rd Ohio State-Denison conference, Granville, OH, USA, May 1996. Boston, MA: Birkhäuser. Trends in Mathematics. 221-238 (1997).
The concept of weakly injective modules was introduced by Jain and López-Permouth and has been widely studied also by many others. For any right $$R$$ module $$M$$, let $$\sigma[M]$$ denote the full subcategory of $$\text{mod-}R$$, consisting of all submodules of $$M$$-generated modules. Let $$E(N)$$ denote the injective hull of a module $$N$$, and let $$A\in\sigma[M]$$. A module $$N\in\sigma[M]$$ is said to be weakly $$A$$-injective in $$\sigma[M]$$ if for any homomorphism $$\varphi\colon A\to E(N)$$, there exists a submodule $$X$$ of $$E(N)$$ such that $$\varphi(A)\subseteq X\cong N$$. $$N$$ is called weakly injective in $$\sigma[M]$$ if $$N$$ is weakly $$A$$-injective for all finitely generated modules $$A$$ in $$\sigma[M]$$. It is well known that every weakly $$A$$-injective module $$N$$ in $$\sigma[M]$$ satisfies the property that for every submodule $$K$$ of $$A$$ if $$A/K$$ embeds in $$E(N)$$ then $$A/K$$ embeds in $$N$$ (a module with this property is called $$A$$-tight in $$\sigma[M]$$). The authors generalize many known results from $$\text{mod-}R$$ to $$\sigma[M]$$. In some of these results, $$M$$ is not assumed to be projective or finitely generated. Some additional conditions are given under which a module which is $$A$$-tight in $$\sigma[M]$$ becomes $$A$$-weakly injective in $$\sigma[M]$$. The authors also study modules $$N$$ in $$\sigma[M]$$ such that every submodule of $$N$$ is weakly injective (tight) in $$\sigma[M]$$. A module is said to be l.f.d. if every finitely generated submodule has finite Goldie dimension. $$\sigma[M]$$ is l.f.d. if every module in $$\sigma[M]$$ is an l.f.d. module. It is shown that $$\sigma[M]$$ is l.f.d. if and only if direct sums of arbitrary collections of weakly injective (tight) modules in $$\sigma[M]$$ are weakly injective (tight) in $$\sigma[M]$$.
For the entire collection see [Zbl 0878.00051].
##### MSC:
 16D50 Injective modules, self-injective associative rings 16D90 Module categories in associative algebras