The Existence of Relative pure Injective Envelopes

Let $\mathcal{S}$ be a class of finitely presented $R$-modules such that $R\in \mathcal{S}$ and $\mathcal{S}$ has a subset $\mathcal{S}^*,$ with the property that for any $U\in \mathcal{S}$ there is a $U^*\in \mathcal{S}^*$ with $U^*\cong U.$ 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.

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