Multiplication semimodules

Rafieh Razavi Nazari; Shaban Ghalandarzadeh

doi:10.2478/ausm-2019-0014

Abstract

Let S be a semiring. An S-semimodule M is called a multiplication semimodule if for each subsemimodule N of M there exists an ideal I of S such that N = IM. In this paper we investigate some properties of multiplication semimodules and generalize some results on multiplication modules to semimodules. We show that every multiplicatively cancellative multiplication semimodule is finitely generated and projective. Moreover, we characterize finitely generated cancellative multiplication S-semimodules when S is a yoked semiring such that every maximal ideal of S is subtractive.

References

[1] D. D. Anderson, Y. Al-Shaniafi, Multiplication modules and the ideal θ(M), Comm. Algebra, 30 (7) (2002), 3383–3390.10.1081/AGB-120004493
Search in Google Scholar Back to article
[2] A. Barnard, Multiplication modules, J. Algebra, 71 (1981), 174–178.10.1016/0021-8693(81)90112-5
Search in Google Scholar Back to article
[3] R. P. Deore, Characterizations of Semimodules, Southeast Asian Bull. Math., 36 (2012), 187–196.
Search in Google Scholar Back to article
[4] R. Ebrahimi Atani, S. Ebrahimi Atani, On subsemimodules of semimodules, Bul. Acad. Ştiint¸e Repub. Mold. Mat., 63 (2) (2010), 20–30.
Search in Google Scholar Back to article
[5] S. Ebrahimi Atani, R. Ebrahimi Atani, U. Tekir, A Zariski topology for semimodules, Eur. J. Pure Appl. Math., 4 (3) (2011), 251–265.10.5373/jarpm.754.012811
Search in Google Scholar Back to article
[6] S. Ebrahimi Atani, M. Shajari Kohan, A note on finitely generated multiplication semimodules over commutative semirings, Int. J. Algebra, 4 (8) (2010), 389–396.
Search in Google Scholar Back to article
[7] Z. A. El-Bast, P. F. Smith, Multiplication modules, Comm. Algebra, 16 (4) (1988), 755–779.10.1080/00927878808823601
Search in Google Scholar Back to article
[8] Sh. Ghalandarzadeh, P. Nasehpour and R. Razavi, Invertible ideals and Gaussian semirings, Arch. Math. (Brno), 53 (2017), 179–192.10.5817/AM2017-3-179
Search in Google Scholar Back to article
[9] J. S. Golan, Semirings and Their Applications, Kluwer Academic Publishers, Dordrecht, 1999.10.1007/978-94-015-9333-5
Search in Google Scholar Back to article
[10] C.B. Kim, A note on the localization in Semirings, J. Sci. Inst. Kook Min Univ, 3 (1985).
Search in Google Scholar Back to article
[11] S. LaGrassa, Semirings: Ideals and Polynomials, PhD Thesis, University of Iowa, 1995.
Search in Google Scholar Back to article
[12] T. Y. Lam, Lectures on Modules and Rings, Springer-Verlag, 1998.10.1007/978-1-4612-0525-8
Search in Google Scholar Back to article
[13] M. D. Larsen, P. J. McCarthy, Multiplicative Theory of Ideals, Academic Press, New York, 1971.
Search in Google Scholar Back to article
[14] A. G. Naoum, A. S. Mijbass, Weak cancellation modules, Kyungpook Math. J., 37 (1997), 73–82.
Search in Google Scholar Back to article
[15] D. G. Northcott, Lessons on Rings, Modules and Multiplicity, New York: Cambridge University Press, 1968.10.1017/CBO9780511565922
Search in Google Scholar Back to article
[16] M. Shabir, Some characterizations and sheaf representations of regular and weakly regular monoids and semirings, PhD Thesis, Quaid-I-Azam University, Pakistan, 1995.
Search in Google Scholar Back to article
[17] P. F. Smith, Some remarks on multiplication modules, Arch. Math. (Basel), 50 (1988), 223–235.10.1007/BF01187738
Search in Google Scholar Back to article
[18] H. A. Tavallaee, M. Zolfaghari, On semiprime subsemimodules and related results, J. Indones. Math. Soc., 19 (2013), 49–59.10.22342/jims.19.1.129.49-59
Search in Google Scholar Back to article
[19] A. Tuganbaev, Multiplication modules, J. Math. Sci., 123 (2) (2004), 3839–3905.10.1023/B:JOTH.0000036653.76231.05
Search in Google Scholar Back to article
[20] G. Yesilot, K. Orel, U. Tekir, On prime subsemimodules of semimodules, Int. J. Algebra, 4 (2010), 53–60.
Search in Google Scholar Back to article

Multiplication semimodules

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