Korean J. Math.  Vol 25, No 4 (2017)  pp.587-606
DOI: http://dx.doi.org/10.11568/kjm.2017.25.4.587

(Co) retractability and (Co) Semi-potency

Hamza Ibrahim Hakmi


This paper is a continuation of study semi-potentness endomorphism rings of module. We give some other characterizations of endomorphism ring to be semi-potent. New results are obtained including necessary and sufficient conditions for the endomorphism ring of semi(injective) projective module to be semi-potent. Finally, we characterize a module $M$ whose endomorphism ring it is semi-potent via direct(injective) projective modules. Several properties of the endomorphism ring of a semi(injective) projective module are obtained. Besides to that, many necessary and sufficient conditions are obtained for semi-projective, semi-injective modules to be semi-potent and co-semi-potent modules.


Semi-potent ring, Semi(injective) projective module, Direct(injective) projective module, (Co)retractable modules, Endomorphism Ring, (Co)Singular ideal.

Subject classification

6E50, 16E70, 16D40, 16D50.


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A. N. Abyzov, I0∗− Modules, Mat. Zametki, 08 (2014), 1–17. (Google Scholar)

B. Amini, Ershad M., and Sharif H, Co-retractable modules, J. Aust. Math. Soc. 86 (3) (2009), 289- 304. (Google Scholar)

A. Haghany and M. R. Vedadi, Study of semi-projective retractable modules, Algebra Colloquium. 14 (3) (2007), 489–496. (Google Scholar)

H. Hamza, I0−Rings and I0−Modules, Math. J. Okayama Univ. 40, (1998), 91–97. (Google Scholar)

F. Kasch and A. Mader, Rings, Modules, and the Total, Front. Math. Birkhauser Verlag. Basel. 2004. (Google Scholar)

W. K. Nicholson, I−Rings, Trans. Amer. Math. Soc. 207 (1975), 361–373. (Google Scholar)

H. Tansee and S. Wongwai, A note on semi-projective modules, Kyungpook Math. 42 (2002), 369–380. (Google Scholar)

A. A. Tuganbaev, Rings over which all modules are I0−modules, Fundam. Prikl. Mat. 13 (2007), 185–194. (Google Scholar)

R. Ware, Endomorphism rings of projective modules, Trans. Amer. Math. Soc. 155 (1971), 233–256. (Google Scholar)

R. Wisbauer, Foundations of Modules and Rings Theory, Philadelphia: Gordon and Breach. 1991. (Google Scholar)


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