INJECTIVE MODULES WITH RESPECT TO MODULES OF PROJECTIVE DIMENSION AT MOST ONE

Several authors have been interested in cotorsion theories. Among these theories we figure the pairs $(\mathcal P_n,\mathcal P_n^{\perp})$, where $\mathcal P_n$ designates the set of modules of projective dimension at most a given integer $n\geq 1$  over a ring $R$. In this paper, we shall focus on homological properties of the class $\mathcal P_1^{\perp}$ that we term the class of $\mathcal P_1$-injective modules. Numerous nice characterizations of rings as well as of their homological dimensions arise from this study. In particular, it is shown that a ring $R$ is left hereditary if and only if any $\mathcal P_1$-injective module is injective and that $R$ is  left semi-hereditary if and only if any $\mathcal P_1$-injective module is FP-injective. Moreover, we prove that the global dimensions of $R$ might be computed in terms of $\mathcal P_1$-injective modules, namely the formula for the global dimension and the weak global dimension turn out to be as follows $$\wdim(R)=\sup \{\fd_R(M): M\mbox { is a }\mathcal P_1\mbox {-injective left } R\mbox {-module} \}$$ and $$\gdim(R)=\sup \{\pd_R(M):M \mbox { is a }\mathcal P_1\mbox {-injective left }R\mbox {-module}\}.$$ We close the paper by proving that, given a Matlis domain $R$ and an $R$-module $M\in\mathcal P_1$, $\Hom_R(M,N)$ is $\mathcal P_1$-injective for each $\mathcal P_1$-injective module $N$ if and only if $M$ is strongly flat.