International Journal of Nonlinear Analysis and Applications

projective modules relative to a semiradical property

Document Type : Research Paper

Authors

Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq

Abstract

In this paper, we introduce a generalization of the projective modules. We show that for a module $M=M_1 \bigoplus M_2$. If $M_2$ is s.p-$M_1$-projective, then for every s.p-closed submodule $A$ of $M$ with $M=M_1+A$, there exists a submodule $K$ of $A$ such that $M=M_1 \bigoplus K$.

Keywords

[1] E. Al–Dhaheri and B. Al–Bahrani, On essential (complement) submodules with respect to an arbitrary submodule, Iraqi J. Sci. 59 (2018), no. 2B, 904–908.
[2] E.A.M. Al-Dhaheri and B.H. Al-Bahrani, Semisimple modules relative to a semiradical property, Iraqi J. Sci. 63 (2022), no. 11, 4901–4910.
[3] G.F. Birkenmeier, B.J. Muller and S.T. Rizvi, Modules in which every fully invariant submodules is essential in a direct Summand, Comm. Algebra 30 (2002), 1395–1415.
[4] T.J. Cheathem and J.R. Smith, Regular modules and Semisimple modules, Pacific J. Math. 65 (1976), 315–323.
[5] J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting modules, supplements and projectivity in module theory, Springer Science & Business Media, 2006.
[6] K. Goodearl, Ring theory: Nonsingular rings and modules, CRC Press, 1976.
[7] N. Hamad and B. AL-Hashimi, Some results on the Jacobson radicals and the M-radicals basic, Sci. Eng. 1 (2002), no. 2A.
[8] F. Kasch, Modules and rings, Academic Press, London, 1982.
[9] S.H. Mohamed and B.J. Muller, Continuous and discrete modules, London Math. Soc. LNS 147 Cambridge Univ. Press, Cambridge, 1990.
[10] A.C. Ozcan, A. Harmanci and P.F. Smith, Duo modules, Glasgow Math. J. 48 (2006), 533–545.
[11] G.V. Wilson, Modules with the direct summand intersection property, Comm. Algebra 14 (1986), no. 1, 21–38.
[12] S.M. Yaseen, On F-regular modules, M.S. thesis, Univ. of Baghdad, Iraq, 1993.
[13] Y. Zhou, Generalization of perfect, semiperfect, and semiregular rings, Algebra Colloq. 7 (2000), no. 3, 305–318.
International Journal of Nonlinear Analysis and Applications
Volume 14, Issue 10
October 2023
Pages 283-292
  • Receive Date: 16 November 2022
  • Revise Date: 19 February 2023
  • Accept Date: 02 March 2023