On $L_{n}$-Injective Modules and $L_{n}$-Injective Dimensions

Abstract

Let $R$ be a ring, and $n$ a fixed nonnegative integer. An $R$-module $W$ is called $L_{n}$-injective if ${\rm Ext}_{R}^{1}(M,W)=0$ for any $R$-module $M$ with flat dimension at most $n$. In this paper, we prove first that ($\mathcal{F}_{n},\mathcal{L}_{n}$) is a complete hereditary cotorsion theory, where $\mathcal{F}_n$ (resp. $\mathcal{L}_n$) denotes the class of all $R$-modules with flat dimension at most $n$ (resp. $L_{n}$-injective $R$-modules). Then we introduce the $L_{n}$-injective dimension of a module and $L_n$-global dimension of a ring. Finally, over rings with weak global dimension $\leq n$, perfect rings, and $L_n$-hereditary rings, more properties and applications of $L_{n}$-injective modules, $L_{n}$-injective dimensions of modules and $\mathcal{L}_{n}$-global dimensions of rings are given.