The Existence of Relative pure Injective Envelopes

Abstract

Let $\mathcal{S}$ be a class of finitely presented $R$-modules such that $R\in \mathcal{S}$ and $\mathcal{S}$ has a subset ${\mathcal{S}}^{\ast },$ with the property that for any $U\in \mathcal{S}$ there is a $U^{\ast }\in {\mathcal{S}}^{\ast }$ with $U^{\ast }\cong U.$ We show that the class of $\mathcal{S}$-pure injective $R$-modules is preenveloping. As an application, we deduce that the left global $\mathcal{S}$-pure projective dimension of $R$ is equal to its left global $\mathcal{S}$-pure injective dimension. As our main result, we prove that, in fact, the class of $\mathcal{S}$-pure injective $R$-modules is enveloping.

to appear in Colloquium Mathematicum