^{}Department of Mathematics, Sinop University, 57000, Sinop, Turkey.

Receive Date: 02 July 2015,
Revise Date: 08 September 2015,
Accept Date: 15 September 2015

Abstract

We call a ring $R$ right generalized semiperfect if every simple right $R$-module is an epimorphic image of a flat right $R$-module with small kernel, that is, every simple right $R$-module has a flat $B$-cover. We give some properties of such rings along with examples. We introduce flat strong covers as flat covers which are also flat $B$-covers and give characterizations of $A$-perfect, $B$-perfect and perfect rings in terms of flat strong covers.

A. Amini, B. Amini, M. Ershad and H. Sharif, On generalized perfect rings, Comm. Algebra 35 (2007), no. 3, 953--963.

A. Amini, M. Ershad and H. Sharif, Rings over which at covers of finitely generated modules are projective, Comm. Algebra 36 (2008), no. 8, 2862--2871.

F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Grad. Texts in Math., 13, Springer-Verlag, New York, 1992.

[4] M. F. Atiyah, and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969.

H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960) 466--488.

E. Buyukask, Rings over which at covers of simple modules are projective, J. Algebra Appl. 11 (2012), no. 3, 1250046, 7 pages.

N. Ding and J. Chen, On a characterization of perfect rings, Comm. Algebra 27 (1999), no. 2, 785--791.

E. E. Enochs, Injective and at covers, envelopes and resolvents, Israel J. Math. 39 (1981), no. 3, 189--209.

F. Kasch, Modules and Rigs, London Math. Soc. Monogr. Ser. 17, Translated from German and edited by D. A. R. Wallace, Academic Press, London- New York, 1982.

T. Y. Lam, Lectures on Modules and Rings, Grad. Texts in Math., 189, Springer-Verlag, New York, 1999.

C. Lomp, On semilocal modules and rings, Comm. Algebra 27 (1999), no. 4, 1921--1935.

A. K. Srivastava, On

A. Amini, B. Amini, M. Ershad and H. Sharif, On generalized perfect rings, Comm. Algebra 35 (2007), no. 3, 953--963.

A. Amini, M. Ershad and H. Sharif, Rings over which at covers of finitely generated modules are projective, Comm. Algebra 36 (2008), no. 8, 2862--2871.

F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Grad. Texts in Math., 13, Springer-Verlag, New York, 1992.

[4] M. F. Atiyah, and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969.

H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960) 466--488.

E. Buyukask, Rings over which at covers of simple modules are projective, J. Algebra Appl. 11 (2012), no. 3, 1250046, 7 pages.

N. Ding and J. Chen, On a characterization of perfect rings, Comm. Algebra 27 (1999), no. 2, 785--791.

E. E. Enochs, Injective and at covers, envelopes and resolvents, Israel J. Math. 39 (1981), no. 3, 189--209.

F. Kasch, Modules and Rigs, London Math. Soc. Monogr. Ser. 17, Translated from German and edited by D. A. R. Wallace, Academic Press, London- New York, 1982.

T. Y. Lam, Lectures on Modules and Rings, Grad. Texts in Math., 189, Springer-Verlag, New York, 1999.

C. Lomp, On semilocal modules and rings, Comm. Algebra 27 (1999), no. 4, 1921--1935.

A. K. Srivastava, On ∑-V rings, Comm. Algebra 39 (2011), no. 7, 2430--2436.

J. Xu, Flat Covers of Modules, Lecture Notes in Math., 1634, Springer-Verlag, Berlin, 1996.