# Wydawnictwa / Czasopisma IMPAN / Colloquium Mathematicum / Wszystkie zeszyty

## The existence of relative pure injective envelopes

### Tom 130 / 2013

Colloquium Mathematicum 130 (2013), 251-264 MSC: Primary 16D70; Secondary 16D10. DOI: 10.4064/cm130-2-7

#### Streszczenie

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.

#### Autorzy

• Fatemeh Zareh-KhoshchehrehDepartment of Mathematics
Alzahra University
Vanak, Post Code 19834
Tehran, Iran
e-mail
• Kamran Divaani-AazarDepartment 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
e-mail

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