Title & Authors
Ozdemir, Salahattin;

Abstract
Let R be a ring, and let M be a left R-module. If M is Rad-supplementing, then every direct summand of M is Rad-supplementing, but not each factor module of M. Any finite direct sum of Rad-supplementing modules is Rad-supplementing. Every module with composition series is (Rad-)supplementing. M has a Rad-supplement in its injective envelope if and only if M has a Rad-supplement in every essential extension. R is left perfect if and only if R is semilocal, reduced and the free left R-module $\small{(_RR)^{({\mathbb{N})}}$ is Rad-supplementing if and only if R is reduced and the free left R-module $\small{(_RR)^{({\mathbb{N})}}$ is ample Rad-supplementing. M is ample Rad-supplementing if and only if every submodule of M is Rad-supplementing. Every left R-module is (ample) Rad-supplementing if and only if R/P(R) is left perfect, where P(R) is the sum of all left ideals I of R such that Rad I = I.
Keywords
Language
English
Cited by
References
1.
K. Al-Takhman, C. Lomp, and R. Wisbauer, ${\tau}$-complemented and ${\tau}$-supplemented modules, Algebra Discrete Math. (2006), no. 3, 1-16.

2.
F.W. Anderson and K. R. Fuller, Rings and Categories of Modules, New-York, Springer, 1992.

3.
J. Averdunk, Moduln mit Erganzungseigenschaft, Dissertation, Ludwig-Maximilians-Universitat Munchen, Fakultat fur Mathematik, 1996.

4.
I. Beck, Projective and free modules, Math. Z. 129 (1972), 231-234.

5.
E. Buyukasik and C. Lomp, Rings whose modules are weakly supplemented are perfect: Applications to certain ring extensions, Math. Scand. 105 (2009), no. 1, 25-30.

6.
E. Buyukasik, E. Mermut, and S. Ozdemir, Rad-supplemented modules, Rend. Semin. Mat. Univ. Padova 124 (2010), 157-177.

7.
E. Buyukasik, R. Tribak, On w-local modules and Rad-supplemented modules, J. Korean Math. Soc. 51 (2014), no. 5, 971-985.

8.
H. Cartan and S. Eilenberg, Homological Algebra, Princeton Landmarks in Mathematics and Physics series, New Jersey: Princeton Univesity, 1956.

9.
J. Clark, C. Lomp, N. Vanaja, and R. Wisbauer, Lifting modules, Frontiers in Mathematics, Basel: Birkhauser Verlag, Supplements and projectivity in module theory, 2006.

10.
E. E. Enochs and O. M. G. Jenda, Relative homological algebra, vol. 30 of de Gruyter Expositions in Mathematics, Berlin: Walter de Gruyter & Co., 2000.

11.
L. Fuchs and L. Salce, Modules over non-Noetherian domains, vol. 84 of Mathematical Surveys and Monographs, Providence, RI: American Mathematical Society, 2001.

12.
F. Kasch and E. A. Mares, Eine Kennzeichnung semi-perfekter Moduln, Nagoya Math. J. 27 (1966), 525-529.

13.
T. Y. Lam, Lectures on modules and rings, vol. 189 of Graduate Texts in Mathematics, New York: Springer-Verlag, 1999.

14.
T. Y. Lam, A first course in noncommutative rings, vol. 131 of Graduate Texts in Mathematics, New York: Springer-Verlag, 2001.

15.
E. Mermut, Homological Approach to Complements and Supplements, Ph.D. thesis, Dokuz Eylul University, The Graduate School of Natural and Applied Sciences, Izmir-Turkey, 2004.

16.
B. L. Osofsky, Rings all of whose finitely generated modules are injective, Pacific J. Math. 14 (1964), 645-650.

17.
J. J. Rotman, An Introduction to Homological Algebra, Universitext, New York: Springer, 2009.

18.
Y.Wang and N. Ding, Generalized supplemented modules, Taiwanese J. Math. 10 (2006), no. 6, 1589-1601.

19.
R. Wisbauer, Foundations of Module and Ring Theory, Reading: Gordon and Breach, 1991.

20.
W. Xue, Characterization of semiperfect and perfect rings, Publ. Mat. 40 (1996), no. 1, 115-125.

21.
H. Zoschinger, Komplementierte Moduln uber Dedekindringen, J. Algebra 29 (1974), 42-56.

22.
H. Zoschinger, Moduln, die in jeder Erweiterung ein Komplement haben, Math. Scand. 35 (1974), 267-287.