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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
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