‏A ‎generalization ‎of‎ ‎‎r‎-submodules ‎

Document Type : Research Paper

Author

Department of Mathematics, University of Payame Noor,Tehran, Iran

Abstract

‎Let $R$ be a commutative ring with identity $1\neq 0$ and $M$ a non-zero unital $R$-module‎. ‎In this paper‎, ‎we present the concept of fully $I$-submodules of $M$ such that $I$ is an ideal of $R$ which is a generalization of $r$-submodules‎. ‎Consider that $I$ is an ideal of $R$‎, ‎a proper submodule $N$ of $M$ is a fully $I$-submodule if $JK\subseteq N$ with ${\rm ann}_{M}(J)=0_{M}$ results that $K\subseteq N$ for each submodule $K$ of $M$ and each ideal $J$ of $R$‎. ‎In addition‎, ‎we present the concept of fully special $I$-submodules which is a generalization of special $r$-submodules‎. ‎A proper submodule $N$ of $M$ is a fully special $I$-submodule if the inclusion $IL\subseteq N$ with ${\rm ann}_{R}(L) = 0_{R}$‎, ‎implies that $I\subseteq (N:M)$ for each submodule $L$ of $M$ and each ideal $J$ of $R$‎. ‎We explore certain outcomes related to these categories of submodules‎.

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References

  1. M. Ahmadi and J. Moghaderi, n-submodules, Iran. J. Math. Sci. Inform., (1) 17 (2022), 177-190.
  2.  H. Ansari-Toroghy and F. Farshadifar, The dual notion of multiplication modules, Taiwan. J. Math., (11) 4 (2007), 1189-1201.
  3.  H. Ansari-Toroghy and F. Farshadifar, Strong comultiplication modules, CMU. J. Nat. Sci. (1) 8 (2009), 105-113.
  4. H. Ansari-Toroghy and F. Farshadifar, On comultiplication modules, Korean Ann. Math., (2008), 1-10.
  5.  Z.El. Bast and P. F. Smith, multiplication module, Comm. Algebra., (4) 16 (1988), 755-779.
  6.  C.Y. Hong, N. K. Kim, Y. Lee and S. J. Ryu, Rings with property (A) and their extensions, J. Algebra, 315 (2007), 612-628.
  7.  I. Kaplansky, Commutative rings, University of Chicago Press, Chicago and London, 1974.
  8. S. Koc and U. Tekir, r-Submodules and sr-Submodules, Turk. J. Math., 42 (2018), 1863-1876.
  9.  R. Mohamadian, r -Ideals in commutative rings, Turk. J. Math., 39 (2015), 733-749.
  10.  R. Mohamadian, r -submodules and uz-modules, Alg. Struc. Appl. (1) 8 (2021), 61-73.
  11. S. Rajaee, S-small and S-essential submodules, J. Algebra Relat. Topics, (1) 10 (2022), 1-10.
  12. S. Rajaee, Essential submodules relative to a submodule, J. Algebra Relat. Topics, (2) 11 (2023), 59-71.
  13. U. Tekir, S. Koc and K. H. Oral, n-ideals of Commutative Rings, Filomat, (10) 31 (2017), 2933-2941.
Journal of Algebra and Related Topics
Volume 12, Issue 2
January 2025
Pages 165-178

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