An approach to extending modules via homomorphisms

Document Type : Research Paper

Author

Department of Mathematics, Quchan University of Technology, P.O.Box 94771-67335, Quchan, Iran

Abstract

The notion of $\mathcal{K}$-extending modules was defined recently as a proper generalization of both extending modules and Rickart modules. Let $M$ be a right $R$-module and let $S=End_R(M)$. We recall that $M$ is a $\mathcal{K}$-extending module if for every element $\phi\in S$, $Ker\phi$ is essential in a direct summand of $M$. Since a direct sum of $\mathcal{K}$-extending modules is not a $\mathcal{K}$-extending module in general, an open question is to find necessary and sufficient conditions for such a direct sum to be $\mathcal{K}$-extending. In this paper, we give an answer to this question. We show that if $M_i$ is $M_j$-injective for all $i, j\in I =\{1, 2, \dots, n\}$, then $\bigoplus_{i=1}^n M_i$ is a $\mathcal{K}$-extending module if and only if $M_i$ is $M_j$-$\mathcal{K}$-extending for all $i, j \in I$. Other results on $\mathcal{K}$-extending modules and some of their applications are also included. 

Keywords

References

[1] A. N. Abyzov and T. H. N. Nhan, CS-Rickart Modules, Lobachevskii J. Math., 35 No. 4 (2014) 317-326.
[2] T. Amouzegar, A Generalization of Lifting Modules, Ukrainian Math. J., 66 No. 11 (2014) 1654-1664.
[3] T. Amouzegar, On K-extending Modules, Tamkang J. Math., 48 No. 1 (2017) 1-11.
[4] T. Amouzegar and A. R. M. Hamzekolaee, Lifting Modules with Respect to Images of a Fully Invariant Submodule, Novi Sad J. Math., 50 No. 2 (2020) 41-50.
[5] K. I. Beidar and F. Kasch, Good Conditions for the Total, International Symposium on Ring Theory (Kyongju, 1999). Trends Mathematics. MA: Birkhäuser, Boston, 43-65, 2001.
[6] N. V. Dung, D. V. Huynh, P. F. Smith and R. Wisbauer, Extending Modules, Pitman Research Notes in Mathematics Series, Longman, Harlow, 1994.
[7] I. Kaplansky, Infinite Abelian Groups, University of Michigan Press, 1969.
[8] F. Karabacak and A. Tercan, On Modules and Matrix Rings with SIP-extending, Taiwanese J. Math., 11 (2007) 1037-1044.
[9] F. Kasch, A. Mader, Rings, Modules and the Total, Basel: Birkhäuser Verlag (Frontiers in Mathematics), 2004.
[10] T. Y. Lam, Lectures on Modules and Rings, GTM 189, Springer Verlag, Berlin-Heidelberg-New York, 1999.
[11] G. Lee, S. T. Rizvi, and C. S. Roman, Rickart modules, Comm. Algebra, 38 No. 11 (2010) 4005-4027.
[12] G. Lee, S. T. Rizvi, and C. S. Roman, Direct Sums of Rickart Modules, J. Algebra, 353 (2012) 62-78.
[13] S. H. Mohamed and B. J. Müller, Continuous and Discrete Modules, London Math. Soc. Lecture Notes Series 147, University Press, Cambridge, 1990.
[14] A. R. Moniri Hamzekolaee and T. Amouzegar, H-supplemented Modules with Respect to Images of Fully Invariant Submodules, Proyecciones (Antofagasta, On line), 40 No. 1 (2021) 35-48.
[15] W. K. Nicholson, Semiregular Modules and Rings, Canad. J. Math., 28 (1976) 1105-1120.
[16] W. K. Nicholson and M. F. Yousif, Weakly Continuous and C2-rings, Comm. Algebra, 29 (2001) 2429-2446.
[17] W. K. Nicholson and Y. Zhou, Semiregular Morphisms, Comm. Algebra, 34 No. 1 (2006) 219-233.
[18] S. K. Sany, A. R. M. Hamzekolaee and Y. Talebi, A Homological Approach to -supplemented Modules, Novi Sad J. Math., 50 No. 2 (2020) 131-142.
[19] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Reading, 1991.
Algebraic Structures and Their Applications
Volume 9, Issue 1
February 2022
Pages 31-39

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