The notions of irreducible ideals of the endomorphism ring on the category of rings and the category of modules

Fitriana Hasnani, Meryta Febrilian Fatimah, Nikken Prima Puspita

Abstract


Let R commutative ring with multiplicative identity, and M is an R-module. An ideal I of R is irreducible if the intersection of every two ideals of R equals I, then one of them is I itself. Module theory is also known as an irreducible submodule, from the concept of an irreducible ideal in the ring. The set of R - module homomorphisms from M to itself is denoted by EndR(M). It is called a module endomorphism M of ring R. The set EndR(M) is also a ring with an addition operation and composition function. This paper showed the sufficient condition of an irreducible ideal on the ring of EndR(R) and EndR(M)


Keywords


Endomorphism; Endomorphism Ring; Irreducible Ideal; Irreducible Submodule

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References


Abdullah, N. K. (2012). Irreducible Submodules and Strongly Irreducible Submodules. Tikrit Journal of Pure Science, 17(4), 219-224.

Adkins, W. A., and Weintraub, S. H. (1992). Algebra: An Approach via Module Theory. New York: Springer-Verlag.

Albu, T., & Smith, P. F. (2009). Primality, Irreducibility, and Complete Irreducibility in Modules Over Commutative Rings. Rev. Roumaine Math. Pures Appl., 54(4), 275-286.

Anderson, F. W., & Fuller, K. R. (1992). Rings and Categories of Modules Second Edition. New York: Springer-Verlag.

Atani, R. E., & Atani, S. E. (2008). Ideal Theory in Commutative Semirings. Buletinul Academiei De Ştiinţe, 57(2), 14-23.

Atani, S. E. (2005). Strongly Irreducible Submodules. Bulletin of the Korean Mathematical Society, 121-131.

Dummit, D. S., and Foote, R. M. (2004). Abstract Algebra Third Edition. United States of America: John Willey and Sons, Inc.

Fraleigh, J. B. (1994). A First Course Abstract Algebra. United States of America: Addison-Wesley Publishing Company, Inc.

Heinzer, W. J., Ratliff, L. J., and Rush, D. E. (2002). Strongly Irreducible Ideals Of A Commutative Ring. Journal Of Pure and Applied Algebra, 166(3), 267-275.

Holmes, R. R. (2008). Abstract Algebra II. Alabama, Amerika Serikat: Auburn University.

Iséki, K. (1956). Ideal Theory of Semiring. Proc. Japan Acad, 32(2), 554-559.

Jimmie, G., and Gilbert, L. (1984). Element Of Modern Algebra. Boston: PWS-Kent Publishing Company.

Khaksari, A., Ershad, M., & Sharif, H. (2006). Strongly Irreducible Submodules of Modules. Acta Mathematica Sinica, English Series, 22(4), 1189-1196.

Malik, D., Mordeson, J., and Sen, M. (2007). Introduction to Abstract Algebra. United States of America: Scientific Word.

Mas'oed, F. (2013). Struktur Aljabar. Palembang: Akademia Permata.

Mostafanasab, H., and Darani, A. Y. (2016). 2-Irreducible and Strongly 2-Irreducible Ideals Of Commutative Rings. Miskolc Mathematical Notes, 17(1), 441-455.

Roman, S. (2008). Advanced Linear Algebra Third Edition. New York: Springer Science+Business Media.

Stenström, B. (2001). Lectures on Rings and Modules. Swedia: Stockholms Universitet.

Wahyuni, S., Wijayanti, I. E., Yuwaningsih, D. A., and Hartanto, A. D. (2016). Teori Ring dan Modul. Yogyakarta: Gadjah Mada University Press.

Wisbauer, R. (1991). Foundations of Module and Ring Theory. Dusseldorf: Gordon and Breach Science Publishers.




DOI: http://dx.doi.org/10.24042/ajpm.v13i1.11139

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