A NEW CLASS OF SMALL SUBMODULES

Document Type : Original Manuscript

Authors

1 Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, Iran.

2 Department of Mathematics‎, ‎Payame Noor University‎, ‎P.O.Box 19395-3697, ‎Tehran‎, ‎Iran.

3 Department of Mathematics‎, ‎Payame Noor University‎, ‎P.O.Box 19395-3697‎, ‎Tehran‎, ‎Iran

Abstract

Let $R$ be a commutative ring with identity $1\neq 0$ and $M$ a nonzero unital $R$-module. In this paper, we introduce a new notion of submodules in $M$, namely $T$-semi-annihilator small submodules of $M$ with respect to an arbitrary submodule $T$ of $M$. A submodule $N$ of $M$ is $T$-semi-annihilator small in $M$ provide that for each submodule $X$ of $M$ with $T\subseteq N+X$ implies that ${\rm Ann}(X)\ll (T:M)$.  In addition, we investigate some results concerning to this new class of submodules. Among various results, we prove that for a faithful finitely generated multiplication module $M$, the submodule $N$ of $M$ is a $T$-semi-annihilator small submodule of $M$ if and only if  $(N:M)$ is a $(T:M)$-semi-annihilator small ideal of $R$. Finally, we explore the properties and the behaviour of this structure under ring homomorphism, localization, direct sums and tensor product of them with a faithfully flat $R$-module.

Keywords

References

 1. H. K. Al-Hurmuzy and B. H. Al-Bahrany, R-Annihilator-small submodules, Iraqi J. Sci., (2016), 129–133.
 
2. T. Amouzegar-Kalati and D. Keskin-Tütüncü, Annihilator-small submodules, Bull. Iranian Math. Soc., 39(6) (2013), 1053–1063.
 
3. F. W. Anderson and K. R. Fuller, Rings and categories of modules, Springer-Verlag Berlin
 
Heidelberg New York, 1992.
 
4. H. Ansari-Toroghy and F. Farshadifar, Strong comultiplication modules, CMU. J. Nat. Sci., 8(1) (2009), 105–113.
 
5. A. Barnard, Multiplication modules, J. Algebra, 71 (1981), 74–178.
 
6. R. Beyranvand and F. Moradi, Small submodules with respect to an arbitrary submodule, J. Algebra Relat. Topics, 3(2) (2015), 43–51.
 
7. Z. A. El-Bast and P. F. Smith, Multiplication modules, Comm. Algebra, 16 (1988), 755– 779.
 
8. F. Farzalipour, S. Rajaee and P. Ghiasvand, Some properties of S-semiannihilator small submodules and S-small submodules with respect to submodule, J. Math., (2024), Article ID: 5547197.
 
9. R. W. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, New York, 1972.
 
10. L. H. Helal and S. M. Yaseen, On semiannihilator supplement submodules, Iraqi J. Sci., (2020), 16–20.
 
11. C. P. Lu, Prime submodules of modules, Comment. Math. Univ. St. Pauli., 33(1) (1984), 61–69.
 
12. R. L. McCasland and M. E. Moore, On radicals of submodules of finitely generated modules, Canad. Math. Bull., 29(1) (1986), 37–39.
 
13. A. Nikseresht and H. Sharif, On comultiplication and r-multiplication modules, J. Algebr. Syst., 2 (2014), 1–19.
 
14. S. Rajaee, S-small and S-essential submodules, J. Algebra Relat. Topics, 10(1) (2022), 1–10.
 
15. A. Tuganbaev, Rings Close to Regular, Kluwer Academic, 2002.
 
16. Y. Wang and Y. Liu, A note on comultiplication modules, Algebra Colloq., 21(1) (2014), 147–150. 
Journal of Algebraic Systems
Volume 13, Issue 3
November 2025
Pages 157-174

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