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 ${\mathrm{Ext}}_R^1(M,W)=0$ 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 $\le n$, 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.

arXiv comment19 pages