Abstract
The classes of FP-injective and weakly quasi-Frobenius rings are investigated. The properties of both classes of rings are closely related to the embedding of finitely presented modules in fp-flat and free modules, respectively. Using these properties, we characterize the classes of coherent CF- and FGF-rings. Moreover, it is proved that the group ring R(G) is FP-injective (weakly quasi-Frobenius, respectively) if and only if the ring R is FP-injective (weakly quasi-Frobenius) and G is locally finite. Bibliography: 15 titles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
G. A. Garkusha and A. I. Generalov, “Duality for categories of finitely presented modules,” Algebra Analiz, 11, 139–152 (1999).
I. Herzog, “The Ziegler spectrum of a locally coherent Grothendieck category,” Proc. London Math. Soc., 74, 503–558 (1997).
R. R. Colby, “Rings which have flat injective modules,” J. Algebra, 35, 239–252 (1975).
G. Garkusha, “Grothendieck categories,” Preprint (http://xxx.lanl.gov) (1999).
B. Stenström, “Coherent rings and FP-injective modules,” J. London Math. Soc., 2, 323–329 (1970).
B. Stenström, Rings of Quotients, Springer-Verlag, New York-Heidelberg (1975).
S. Jain, “Flat and FP-injectivity,” Proc. Amer. Math. Soc., 41, 437–442 (1973).
C. Faith, Algebra: Rings, Modules, and Categories. I, Springer-Verlag, New York-Heidelberg (1973).
F. Kasch, Moduln und Ringe, B. G. Teubner, Stuttgart (1977).
M. Prest, P. Rothmaler, and M. Ziegler, “Absolutely pure and flat modules and `indiscrete' rings,” J. Algebra, 174, 349–372 (1995).
C. Faith, Algebra. II. Ring Theory, Springer-Verlag, Berlin-New York (1976).
J. Rada and M. Saorin, “On two problems about embedding of modules in free modules,” Notas de Matemática (Serie: Pre-Print), 170 (1998).
G. Renault, “Sur les anneaux de groupes,” C. R. Acad. Sci. Paris Ser. A, 273, No. 2, 84–87 (1971).
I. G. Connell, “On the group ring,” Can. J. Math., 15, 650–685 (1963).
M.-P. Malliavin, “Sur les anneaux de groupes FP-self-injectifs,” C. R. Acad. Sci. Paris Ser. A, 273, No. 2, 88–91 (1971).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Garkusha, G.A. FP-Injective and Weakly Quasi-Frobenius Rings. Journal of Mathematical Sciences 112, 4303–4312 (2002). https://doi.org/10.1023/A:1020334701055
Issue Date:
DOI: https://doi.org/10.1023/A:1020334701055