Abstract
Let R be a commutative semiring with \(1\ne 0\), M an R-semimodule, and \(n \ge 1\) a positive integer. A proper subsemimodule P of M is called n-absorbing, if whenever \(r_1,\ldots ,r_n \in R\) and \(x \in M\) together with \(r_1 \ldots r_nx \in P\), imply \(r_1 \ldots r_n \in (P : M)\) or there exists \(1 \le i \le n\) such that \(r_1 \ldots \widehat{r_i} \ldots r_nx \in P\). In this paper, we study the concept of n-absorbing subsemimodules that is a generalization of prime subsemimodules. Some basic properties of these subsemimodules and some useful examples of them are investigated. Also, we study the stability of n-absorbing subsemimodules with respect to some various constructions of semimodules.
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References
Ameri, R.: On the prime submodules of multiplication modules. IJMMS 27, 1715–1724 (2003)
Anderson, D.F., Badawi, A.: On \(n\)-absorbing ideals of commutative rings. Commun. Algebra 39, 1646–1672 (2011)
Anderson, D.F., Chapman, S.T.: How far is an element from being prime? J. Algebra Appl. 9, 779–789 (2010)
Atani, S.E.: On primal and weakly primal ideals over commutative semirings. Glas. Mat. 43(63), 13–23 (2008)
Atani, R.E., Atani, S.E.: On subsemimodules of semimodules. Bul. Academiei Stiin. Rep. Mold. Mat. 63(2), 20–30 (2010)
Badawi, A.: On 2-absorbing ideals of commutative rings. Bull. Austral. Math. Soc. 75, 417–429 (2007)
Darani, A.Y., Soheilnia, F.: On \(n\)-absorbing submodules. Math. Commun. 17, 547–557 (2012)
Dubey, M.K., Aggarwal, P.: On \(n\)-absorbing submodules of modules over commutative rings. Beitr. Algebra Geom. (2015). https://doi.org/10.1007/s13366-015-0263-5
Ebrahimpour, M., Nekooei, R.: On generalizations of prime ideals. Commun. Algebra 40, 1268–1279 (2012)
Ebrahimpour, M., Nekooei, R.: On generalizations of prime submodules. Bull. Iran. Math. Soc. 39(5), 919–939 (2013)
Ebrahimpour, M.: On generalizations of prime ideals (II). Commun. Algebra 42, 3861–3875 (2014)
Ebrahimpour, M.: On generalisations of almost prime and weakly pime ideals. Bull. Iran. Math. Soc. 40(2), 531–540 (2014)
Ebrahimpour, M.: Some remarks on generalizations of multiplicatively-closed subsets. J. Algebra Syst. 4(1), 15–27 (2016)
Ebrahimpour, M.: (n-1, n)-weakly prime submodules in direct product of modules. J. Hyperstruct. 6(1), 10–16 (2017)
El-Bast, Z.A., Smith, P.F.: Multiplication modules. Commun. Algebra 16(4), 755–779 (1988)
Golan, J.S.: Semirings and Their Applications. Kluwer Academic Publishers, Dordrecht (1999)
Golan, J.S.: The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science (Pitman Monographs and Surveys in Pure and Applied Mathematics). Longman Scientific and Thechnical, Harlow (1992)
Hebisch, U., Weinert, H.J.: Halbringe Algebraische theorie und anwendungen in der informatik. Teubner Studienbücher, Stuttgart (1993)
Nekooei, R.: Weakly prime submodules. FJMS 39(2), 185–192 (2010)
Yesilot, G.: On prime and maximal \(k\)-subsemimodules of semimodules. Hacettepe J. Math. Stat. 39(3), 305–312 (2010)
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The author gratefully thanks Professor André Leroy for his kind suggestions which significantly improved this paper.
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Ebrahimpour, M. Some remarks on n-absorbing subsemimodules. AAECC 32, 283–297 (2021). https://doi.org/10.1007/s00200-020-00483-3
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DOI: https://doi.org/10.1007/s00200-020-00483-3