Modules whose surjective endomorphisms have a $\gamma$-small kernels

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Sciences, University of Mohammed First, Oujda, Morocco

2 Department of Mathematics, Faculty of Sciences, University of Mohammed First, Oujda, Morocco.

10.29252/as.2022.2677

Abstract

In this paper, we introduce a proper generalization of that of Hopfian modules, called $\gamma$-Hopfian modules. A right $R$-module $M$ is said to be $\gamma$-Hopfian, if any surjective endomorphism of $M$ has a $\gamma$-small kernel. Some basic characterizations of $\gamma$-Hopfian modules are proved. We prove that a ring $R$ is semisimple cosingular if and only if every $R$-module is $\gamma$-Hopfian. Further, we prove that the $\gamma$-Hopfian property is preserved under Morita equivalences. Some other properties of $\gamma$-Hopfian modules are also obtained with examples.

Keywords


[1] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Vol. 13, Springer-Verlag, New York, 1992.
[2] E. P. Armendariz, J. W. Fisher and R. L. Snider, On injective and surective endomorphisms of finitely generated modules, Comm. Alg., 6 No. 7 (1978) 659-672.
[3] P. Aydogdu and A. C. Ozcan, Semi co-Hopfian and Semi Hopfian Modules, East West J. Math., 10 No. 1 (2008) 57-72.
[4] A. El Moussaouy and M. Ziane, Modules in which every surjective endomorphism has a μ-small kernel, Ann. Univ. Ferrara, 66 (2020) 325-337.
[5] A. Ghorbani, A. Haghany, Generalized Hopfian modules, J. Algebra, 255 No. 2 (2002) 324-341.
[6] A. Haghany and M. R. Vedadi, Modules whose injective endomorphisms are essential, J. Algebra, 243 No. 2 (2001) 765-779.
[7] V. A. Hiremath, Hopfian rings and Hopfian modules, Indian J. Pure Appl. Math., 17 No. 7 (1986) 895-900.
[8] M. Hosseinpour and A. R. M. Hamzekolaee, γ-Small submodules and γ-Lifting modules, East West J. Math., 22 No. 1 (2020) 52-63.
[9] A. Koehler, Quasi-projective and quasi-injective modules, Pacific J. Math., 36 No. 3 (1971) 713-720.
[10] A. C. Ozcan and A. Harmanci, Characterizations of Some Rings by Functor Z*(.), Turkish J. Math., 21 (1997) 325-331.
[11] A. C. Ozcan, Modules with small cyclic submodules in their injective hulls, comm. In Algebra., 30 No. 4 (2002) 1575-1589.
[12] K. Varadarajan, Hopfian and co-Hopfian objects. Publications, Matematiques, 36 (1992) 293-317.
[13] K. Wasan and M. K. Enas, On a Generalization of small submodules, Sci. Int. (Lahore), 30 No. 3 (2018) 359-365.