The existence of relative pure injective envelopes

Fatemeh Zareh-Khoshchehreh ¹   ; Kamran Divaani-Aazar ²

¹ Department of Mathematics Alzahra University Vanak, Post Code 19834 Tehran, Iran
² Department of Mathematics Alzahra University Vanak, Post Code 19834 Tehran, Iran and Institute for Studies in Theoretical Physics and Mathematics P.O. Box 19395-5746 Tehran, Iran

Colloquium Mathematicum, Tome 130 (2013) no. 2, pp. 251-264

Résumé

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.

Détail

DOI : 10.4064/cm130-2-7

Keywords: mathcal class finitely presented r modules mathcal mathcal has subset mathcal * property mathcal there * mathcal * * cong class mathcal pure injective r modules preenveloping application deduce global mathcal pure projective dimension equal its global mathcal pure injective dimension main result prove class mathcal pure injective r modules enveloping