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E-small essential submodules | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 73، دوره 13، شماره 1، خرداد 2022، صفحه 881-887 اصل مقاله (340.03 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2022.5608 | ||
| نویسندگان | ||
| Mamoon F. Khalf* 1؛ Hind Fadhil Abbas2 | ||
| 1Department of Physics, College of Education, University of Samarra, Iraq | ||
| 2Directorate of Education Salah Eddin, Khaled Ibn Al Walid School, Tikrit, Iraq | ||
| تاریخ دریافت: 18 مرداد 1400، تاریخ بازنگری: 14 شهریور 1400، تاریخ پذیرش: 02 مهر 1400 | ||
| چکیده | ||
| Let $R$ be a commutative ring with identity, and \(U_{R}\) be an $R$-module, with \(E = End(U_{R})\). In this work we consider a generalization of class small essential submodules namely E-small essential submodules. Where the submodule $Q$ of \(U_{R}\) is said E-small essential if $Q$ \(\cap W = 0\) , when W is a small submodule of \(U_{R}\), implies that \(N_{S}\left( W \right) = 0\), where \(N_{S}\left( W \right) = \left\{ \psi \in E\ |\ Im\psi \subseteq W \right\}\). The intersection \({\overline{B}}_{R}(U)\) of each submodule of \(U_{R}\) contained in \(Soc(U_{R})\). The \({\overline{B}}_{R}(U)\) is unique largest E-small essential submodule of \(U_{R}\), if \(U_{R}\) is cyclic. Also in this paper we study \({\overline{B}}_{R}(U)\) and \({\overline{W}}_{E}\left( U \right)\). The condition when \({\overline{B}}_{R}(U)\) is E-small essential, and \(\text{Tot}\left( \ U,U \right) = {\overline{W}}_{E}\left( U \right) = J(E)\) are given. | ||
| کلیدواژه‌ها | ||
| Small submodule؛ Small essential submodules؛ E-small essential submodules؛ Endomorphism ring | ||
| مراجع | ||
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[1] F.W. Anderson and K.R. Fuller, Rings and Categories of Modules, Springer-Verlag, 1992. [2] J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting Modules, Front. Mathematics, Birk¨auser Verlag, 2006. [3] A. Haghany and M.R. Vedadi, Study of semi-projective retractable modules, Algebra Colloq. 14 (207) 489–496. [4] T. A. Kalati and D.K. T¨ut¨unc¨u, Annihilator-small submodules, Bull. Iran Math. Soc. 39 (2013) 1053–1063. [5] W. K. Nicholson and Y. Zhou, Annihilator-small right ideals, Algebra Colloq. 18 (2011) 785–800. [6] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Reading, 1991. [7] D.X. Zhan and X.R. Zhang, Small-Essential Submodule and Morita Duality, Southeast Asian Bull. Math. 35 (2021) 1051–1062. | ||
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