Rings over which every module has a flat \d-cover

Let M be a module. A d-cover of M is an epimorphism from a module F onto M with a d-small kernel. A d-cover is said to be a flat d-cover in case F is a flat module. In the present paper, we investigate some properties of (flat) d-covers and flat modules having a projective d-cover. Moreover, we study rings over which every module has a flat d-cover and call them right generalized d-perfect rings. We also give some characterizations of d-semiperfect and d-perfect rings in terms of locally (finitely, quasi-, direct-) projective d-covers and flat d-covers.

Rings over which every module has a flat \d-cover

Keywords:

Key words: d-covers, d-perfect rings, d-semiperfect rings flat modules,

PDF

___

Alkan, M., Nicholson, W. K., ¨ Ozcan, A. C ¸ .: A generalization of projective covers. J. Algebra. 319, 4947–4960 (2008). Amini, A., Amini, B., Ershad, M., Sharif, H.: On generalized perfect rings. Comm. Algebra. 35, 953–963 (2007).

Anderson, F. W., Fuller, K. R.: Rings and Categories of Modules. New-York: Springer-Verlag 1992.

Azumaya, G.: Strongly π -regular rings. J. Fac. Sci. Hokkaido Univ. 13, 34–39 (1954).

Azumaya, G.: Some characterizations of regular modules. Publ. Mat. 34, 241–248 (1990).

Bican L., El Bashir R., Enochs E.: All modules have ﬂat covers. Bull. London Math. Soc. 33, 385–390 (2001).

B¨ uy¨ uka¸sık, E., Lomp, C.: When δ -semiperfect rings are semiperfect. Turkish J. Math. 33, 1–8 (2009).

Ding, N., Chen, J.: On a characterization of perfect rings. Comm. Algebra. 27, 785–791 (1999).

Enochs, E., Jenda, O.M.G.: Relative Homological Algebra. Berlin-New York: Walter de Gruyter 2000.

Hamsher, R. M.: Commutative rings over which every module has a maximal submodule. Proc. Amer. Math. Soc. 18, 1133–1137 (1967).

Hausen, J.: Direct projective modules. Bull. Inst. Math. Acad. Sinica. 9, 447–451 (1981).

Kasch, F.: Modules and rings. London-New York: London Mathematical Society Monographs, Academic Press 19 Lam, T. Y.: Lectures on Modules and Rings. New York: Springer-Verlag 1998.

Lomp, C.: On semilocal modules and rings. Comm. Algebra. 27, 1921–1935 (1999).

Nicholson, W. K., Yousif, M. F.: Quasi–Frobenius Rings. Cambridge: Cambridge University Press 2003.

Oneto R., Angel V.: Purity and direct summands. Divulg. Mat. 4, 49–53 (1996).

Yousif, M. F., Zhou, Y.: Semiregular, semiperfect and perfect rings relative to an ideal. Proceedings of the Second Honolulu Conference on Abelian Groups and Modules. Rocky Mountain J. Math. 32, 1651–1671 (2002).

Zhou, Y.: Generalizations of perfect, semiperfect and semiregular rings. Algebra Colloq. 7, 305–318 (2000).

Zimmermann-Huisgen, B.: Pure submodules of direct products of free modules. Math. Ann. 224, 233–245 (1976).