The existence of relative pure injective envelopes
Volume 130 / 2013
Colloquium Mathematicum 130 (2013), 251-264
MSC: Primary 16D70; Secondary 16D10.
DOI: 10.4064/cm130-2-7
Abstract
Let $\mathcal {S}$ be a class of finitely presented $R$-modules such that $R\in \mathcal {S}$ and $\mathcal {S}$ has a subset $\mathcal {S}^*$ 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.
