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 (Pn, P⊥ n ), where Pn designates the set of modules of projective dimension at most a given integer n ≥ 1 over a ring R. In this paper, we shall focus on homological properties of the class P⊥ 1 that we term the class of P1-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 P1- injective module is injective and that R is left semi-hereditary if and only if any P1-injective module is FP-injective. Moreover, we prove that the global dimensions of R might be computed in terms of P1-injective modules, namely the formula for the global dimension and the weak global dimension turn out to be as follows wgl-dim(R) = sup{fdR(M) : M is a P1-injective left R-module} and l-gl-dim(R) = sup{pdR(M) : M is a P1-injective left R-module}. We close the paper by proving that, given a Matlis domain R and an R-module M ∈ P1, HomR(M, N) is P1-injective for each P1-injective module N if and only if M is strongly flat.