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Iraq Al Diwaniyah
Section Mathematics
Title On $\mathcal{L}$-injective modules
Author(-s) Mehdi A.R.a
Affiliations University of Al-Qadisiyaha
Abstract Let $\mathcal{M}=\{(M,N,f,Q)\mid M,N,Q\in R\text{-Mod}, \,N\leq M,\,f\in \text{Hom}_{R}(N,Q)\}$ and let $\mathcal{L}$ be a nonempty subclass of $\mathcal{M}.$ Jirásko introduced the concept of $\mathcal{L}$-injective module as a generalization of injective module as follows: a module $Q$ is said to be $\mathcal{L}$-injective if for each $(B,A,f,Q)\in \mathcal{L}$ there exists a homomorphism $g\colon B\rightarrow Q$ such that $g(a)=f(a),$ for all $a\in A$. The aim of this paper is to study $\mathcal{L}$-injective modules and some related concepts. Some characterizations of $\mathcal{L}$-injective modules are given. We present a version of Baer's criterion for $\mathcal{L}$-injectivity. The concepts of $\mathcal{L}$-$M$-injective module and $s$-$\mathcal{L}$-$M$-injective module are introduced as generalizations of $M$-injective modules and give some results about them. Our version of the generalized Fuchs criterion is given. We obtain conditions under which the class of $\mathcal{L}$-injective modules is closed under direct sums. Finally, we introduce and study the concept of $\sum$-$\mathcal{L}$-injectivity as a generalization of $\sum$-injectivity and $\sum$-$\tau$-injectivity.
Keywords injective module, generalized Fuchs criterion, hereditary torsion theory, $t$-dense, preradical, natural class
UDC 512.553.3
MSC 16D50, 16D10, 16S90
DOI 10.20537/vm180204
Received 3 February 2018
Language English
Citation Mehdi A.R. On $\mathcal{L}$-injective modules, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 2, pp. 176-192.
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