For any module M over an associative ring R, let σ[M] denote the smallest Grothendieck subcategory of Mod-R containing M. If σ[M] is locally finitely presented the notions of purity and pure injectivity are defined in σ[M]. In this paper the relationship between these notions and the corresponding notions defined in Mod-R is investigated, and the connection between the resulting Ziegler spectra is discussed. An example is given of an M such that σ[M] does not contain any non-zero finitely presented objects.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm99-2-4,
author = {Mike Prest and Robert Wisbauer},
title = {Finite presentation and purity in categories $\sigma$[M]},
journal = {Colloquium Mathematicae},
volume = {100},
year = {2004},
pages = {189-202},
zbl = {1061.16009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm99-2-4}
}
Mike Prest; Robert Wisbauer. Finite presentation and purity in categories σ[M]. Colloquium Mathematicae, Tome 100 (2004) pp. 189-202. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm99-2-4/