A new study on semisimple modules applying a fixed submodule

Articles in Press

Document Type : Research Paper

Author

Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

10.22034/as.2026.24452.1877

Abstract

Let $M$ be an $R$-module and $T$ a fixed submodule of $M$. Then we say that $M$ is a $T$-semisimple module if each submodule of $M$ is a $T$-direct summand of $M$, i.e. for each submodule $N$ of $M$, there exists a submodule $K$ of $M$ such that $M=N+K$ and $N\cap K$ is contained in $T$. We also present the definition of a $T$-simple module in terms of simple modules. It is shown that a module $M$ is $T$-semisimple if and only if $M/T$ is a semisimple module. We prove that any homomorphic image and each submodule of a $T$-semisimple module inherits the property with a suitable submodule. As expected, it is proved that $M$ is $T$-semisimple if and only if $M$ is a sum of $T$-simple modules.

Keywords

Main Subjects

References

[1] S. Asgari, Y. Talebi and A. R. Moniri Hamzekolaee, A new approach to (dual) Rickart modules via isomorphisms, Analele Stiintifice ale Universitatii Ovidius Constanta, 32 No. 2 (2024) 5-18.
[2] M. Behboodi, A. Daneshvar and M. R. Vedadi, Virtually semisimple modules and a generalization of the Wedderburn-Artin theorem, Comm. Algebra, 46 No. 6 (2017) 2384-2395.
[3] J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting Modules. Supplements and Projectivity in Module Theory, Frontiers in Mathematics, Birkhäuser, Boston, 2006.
[4] N. V. Ding, D. V. Huynh, P. F. Smith and R. Wisbauer, Extending Modules, Pitman Research Notes in Mathematics Series 313, Longman Scientific, Harlow, 1996.
[5] S. H. Mohamed and B. J. Müller, Continuous and Discrete Modules, London Mathematical Society Lecture Notes Series 147, Cambridge University Press, Cambridge, 1990.
[6] A. R. Moniri Hamzekolaee, H∗-condition on the set of submodules of a module, Alg. Struc. Appl., 6 No. 2 (2019) 13-20.
[7] A. R. Moniri Hamzekolaee, H-supplemented modules and singularity, Alg. Struc. Appl., 7 No. 1 (2020) 49-57.
[8] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Reading, 1991.
Algebraic Structures and Their Applications

Articles in Press, Corrected Proof
Available Online from 07 October 2026

Files

Share

How to cite

Statistics