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MATHEMATICS
On $\mathcal{L}$-injective modules
A. R. Mehdi Department of Mathematics, College of Education,
University of Al-Qadisiyah, Al-Qadisiyah, Iraq
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 Received: 03.02.2018
Language: English
DOI:
10.20537/vm180204