Let Fl(R) denote the category of flat right modules over an associative ring R. We find necessary and sufficient conditions for Fl(R) to be a Grothendieck category, in terms of properties of the ring R.
@article{bwmeta1.element.bwnjournal-article-fmv147i1p83bwm, author = {J. Garc\'\i a and J. Mart\'\i nez Hern\'andez}, title = {When is the category of flat modules abelian?}, journal = {Fundamenta Mathematicae}, volume = {146}, year = {1995}, pages = {83-91}, zbl = {0843.16002}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv147i1p83bwm} }
García, J.; Martínez Hernández, J. When is the category of flat modules abelian?. Fundamenta Mathematicae, Tome 146 (1995) pp. 83-91. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv147i1p83bwm/
[00000] [1] J. Asensio Mayor and J. Martínez Hernández, On flat and projective envelopes, J. Algebra 160 (1993), 434-440. | Zbl 0802.16003
[00001] [2] M. Auslander, Large modules over artin algebras, in: Algebra, Topology and Category Theory, Academic Press, 1976, 3-17.
[00002] [3] R. R. Colby, Rings which have flat injective modules, J. Algebra 35 (1975), 239-252. | Zbl 0306.16015
[00003] [4] E. E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), 189-209. | Zbl 0464.16019
[00004] [5] K. R. Fuller, On rings whose left modules are direct sums of finitely generated modules, Proc. Amer. Math. Soc. 54 (1976), 39-44. | Zbl 0325.16024
[00005] [6] J. L. García and J. Martínez Hernández, Purity through Gabriel's functor rings, Bull. Soc. Math. Belgique 45 (1993), 137-152. | Zbl 0804.18008
[00006] [7] J. Gómez Torrecillas, Anillos con módulos planos libres de torsión, Ph.D. Thesis, University of Granada, 1992.
[00007] [8] J. Gómez Torrecillas and B. Torrecillas, Flat torsionfree modules and QF-3 rings, Osaka J. Math. 30 (1993), 529-542. | Zbl 0803.16031
[00008] [9] M. Hoshino, On dominant dimension of Noetherian rings, Osaka J. Math. 26 (1989), 275-280. | Zbl 0701.16008
[00009] [10] S. Jøndrup and D. Simson, Indecomposable modules over semiperfect rings, J. Algebra 73 (1981), 23-29. | Zbl 0496.16033
[00010] [11] M. F. Jones, Coherence relative to an hereditary torsion theory, Comm. Algebra 10 (1982), 719-739. | Zbl 0483.16027
[00011] [12] R. W. Miller and M. L. Teply, On flatness relative to a torsion theory, ibid. 6 (1978), 1037-1071. | Zbl 0381.16012
[00012] [13] A. Rosenberg and D. Zelinsky, Finiteness of the injective hull, Math. Z. 70 (1959), 372-380. | Zbl 0084.26505
[00013] [14] D. Simson, Functor categories in which every flat object is projective, Bull. Acad. Polon. Sci. 22 (1974), 375-380. | Zbl 0328.18005
[00014] [15] D. Simson, On pure global dimension of locally finitely presented Grothendieck categories, Fund. Math. 96 (1977), 91-116. | Zbl 0361.18010
[00015] [16] B. Stenström, Rings of Quotients, Springer, Berlin, 1975.
[00016] [17] H. Tachikawa, QF-3 rings and categories of projective modules, J. Algebra 28 (1974), 408-413. | Zbl 0281.16009
[00017] [18] W. V. Vasconcelos, The Rings of Dimension Two, Marcel Dekker, New York, 1975. | Zbl 0324.13013