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Essential submodules relative to a submodule | ||
| Journal of Algebra and Related Topics | ||
| دوره 11، شماره 2، اسفند 2023، صفحه 59-71 اصل مقاله (312.46 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2023.23331.1465 | ||
| نویسنده | ||
| S. Rajaee* | ||
| Department of Mathematics, University of Payame Noor,Tehran, Iran. | ||
| چکیده | ||
| In this paper, our aim is to introduce and study the essential submodules of an $R$-module $M$ relative to an arbitrary submodule $T$ of $M$. Let $T$ be an arbitrary submodule of an $R$-module $M$, then we say that a submodule $N$ of $M$ is an essential submodule of $M$ relative to $T$, whenever for every submodule $X$ of $M$, $N\cap X\subseteq T$ implies that $(T:M)\subseteq ^{e}{\rm Ann}(X)$. We investigate some new results concerning to this class of submodules. Among various results we prove that for a faithful multiplication $R$-module $M$, if the submodule $N$ of $M$ is an essential submodule of $M$ relative to $T$, then $(N:M)$ is an essential ideal of $R$ relative to $(T:M)$. The converse is true if $M$ is moreover a finitely generated module. | ||
| کلیدواژه‌ها | ||
| Multiplication module؛ faithful module, essential submodule | ||
| مراجع | ||
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[1] F. W. Anderson and K. R. Fuller, Rings and categories of modules, Springer-Verlag Berlin Heidelberg New York, 1992. [2] H. Ansari-Torogh and F. Farshadifar , Strong comultiplication modules, CMU. J. Nat. Sci. (1) 8 (2009), 105-113. [3] M. F. Atiyah, , I. G. Macdonald , Introduction to commutative algebra, Addison- Wesley, 1969. [4] A.Barnard, Multiplication modules, J. Algebra, 71 (1981), 74-178. [5] Z.A. El-Bast and P.F. Smith, Multiplication modules, Comm. Algebra, 16 (1988) 755-779. [6] K.R. Goodearl and R. B. War eld , An introduction to Noncommutative Noetherian Rings, Cambridge University Press, 2004. [7] B. W. Green, and L. Van Wyk, On the small and essential ideals in certain classes of rings, J. Austral. Math. Soc. Ser. A, 46 (1989), 262 -271. [8] T. Y. Lam, Lectures on Modules and Rings, Springer, 1999. [9] S. Safaeeyan and N. Saboori Shirazi, Essential submodules with respect to an arbitrary submodule, Journal of Mathematical Extension, (3) 7 (2013), 15-27. [10] S. Rajaee, Some results on the quotient of co-m modules, J. Algebra Relat. Topics, 9 (1) (2021), 79-92. [11] S. Rajaee, S-small and S-essential submodules, J. Algebra Relat. Topics, 10 (1) (2022), 1-10. [12] A. Tuganbaev, Rings Close to Regular, Kluwer Academic, 2002. [13] F. Wang and H. Kim, Foundations of Commutative Rings and Their Modules, 2016. | ||
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