On $\mathcal{L}$-injective modules

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.