Finite presentation and purity in categories $\sigma [M]$
Volume 99 / 2004
Colloquium Mathematicum 99 (2004), 189-202
MSC: Primary 16D90; Secondary 18E15.
DOI: 10.4064/cm99-2-4
Abstract
For any module $M$ over an associative ring $R$, let $ \sigma [M] $ denote the smallest Grothendieck subcategory of ${\rm Mod}\hbox {-}R$ containing $M$. If $ \sigma [M]$ is locally finitely presented the notions of purity and pure injectivity are defined in $ \sigma [M]$. In this paper the relationship between these notions and the corresponding notions defined in ${\rm Mod}\hbox {-}R$ is investigated, and the connection between the resulting Ziegler spectra is discussed. An example is given of an $M$ such that $ \sigma [M]$ does not contain any non-zero finitely presented objects.
