Let (????1[[????1, ?1, ????1]]) and ????????(????2[[????2, ?2, ????2]]) be a matrix rings over skew generalized power series rings, where ????1, ????2 are commutative rings with an identity element, (????1, ?1), (????2, ?2) are strictly ordered monoids, ????1: ????1 ? ????????????(????1), ????2: ????2 ? ????????????(????2) are monoid homomorphisms. In this research, we define a mapping ???? from (????1[[????1, ?1, ????1]]) to ????????(????2[[????2, ?2, ????2]]) by using a strictly ordered monoid homomorphism ????: (????1, ?1) ? (????2, ?2), and ring homomorphisms ????: ????1 ? ????2 and ????: ????1[[????1, ?1, ????1]] ? ????2[[????2, ?2, ????2]]. Furthermore, we prove that ???? is a ring homomorphism, and also we give the sufficient conditions for ???? to be a monomorphism, epimorphism, and isomorphism.
Introduction
Conclusion
A ring homomorphism ???? from the matrix ring (????1[[????1, ?1, ????1]]) to the matrix ring ????????(????2[[????2, ?2, ????2]]) can be constructed by using a strictly ordered monoid homomorphism ????: (????1, ?1) ? (????2, ?2), and ring homomorphisms ????: ????1 ? ????2 and ????: ????1[[????1, ?1, ????1]] ? ????2[[????2, ?2, ????2]]. Furthermore, it also proves that Ker(????) is equal to the matrix ring over SGPSR (Ker(????))[[????1, ?1, ????1]]. Moreover, if ???? is an isomorphism and ???? is a monomorphism, then ???? is a monomorphism. While, if ???? is an epimorphism, then ???? is an epimorphism. Consequently, ???? is an isomorphism if ???? is an isomorphism, ???? is a monomorphism, and ???? is an epimorphism.
References
[1] H. Anton and C. Rorres, Elementary Linear Algebra: Applications Version, 9th Edition. New Jersey, 2005.
[2] W. C. Brown, Matrices Over Commutative Rings. New York: Marcel Dekker Inc., 1993.
[3] D. S. Dummit and R. M. Foote, Abstract Algebra, Third Edit. John Wiley and Sons, Inc., 2004.
[4] R. Mazurek and M. Ziembowski, “Uniserial rings of skew generalized power series,” J. Algebr., vol. 318, no. 2, pp. 737–764, 2007.
[5] R. Mazurek and M. Ziembowski, “On von Neumann regular rings of skew generalized power series,” Commun. Algebr., vol. 36, no. 5, pp. 1855–1868, 2008.
[6] R. Mazurek and M. Ziembowski, “The ascending chain condition for principal left or right ideals of skew generalized power series rings,” J. Algebr., vol. 322, no. 4, pp. 983–994, 2009.
[7] R. Mazurek and M. Ziembowski, “Weak dimension and right distributivity of skew generalized power series rings,” J. Math. Soc. Japan, vol. 62, no. 4, pp. 1093–1112, 2010.
[8] R. Mazurek, “Rota-Baxter operators on skew generalized power series rings,” J. Algebr. its Appl., vol. 13, no. 7, pp. 1–10, 2014.
[9] R. Mazurek, “Left principally quasi-Baer and left APP-rings of skew generalized power series,” J. Algebr. its Appl., vol. 14, no. 3, pp. 1–36, 2015.
[10] R. Mazurek and K. Paykan, “Simplicity of skew generalized power series rings,” New York J. Math., vol. 23, pp. 1273–1293, 2017.
[11] A. Faisol, “Homomorfisam Ring Deret Pangkat Teritlak Miring,” J. Sains MIPA, vol. 15, no. 2, pp. 119–124, 2009.
[12] A. Faisol, “Pembentukan Ring Faktor Pada Ring Deret Pangkat Teritlak Miring,” in Prosiding Semirata FMIPA Univerisitas Lampung, 2013, pp. 1–5.
[13] A. Faisol, “Endomorfisma Rigid dan Compatible pada Ring Deret Pangkat Tergeneralisasi Miring,” J. Mat., vol. 17, no. 2, pp. 45–49, 2014.
[14] A. Faisol, B. Surodjo, and S. Wahyuni, “Modul Deret Pangkat Tergeneralisasi Skew T-Noether,” in Prosiding Seminar Nasional Aljabar, Penerapan dan Pembelajarannya, 2016, pp. 95–100.
[15] A. Faisol, B. Surodjo, and S. Wahyuni, “The Impact of the Monoid Homomorphism on The Structure of Skew Generalized Power Series Rings,” Far East J. Math. Sci., vol. 103, no. 7, pp. 1215–1227, 2018.
[16] A. Faisol and Fitriani, “The Sufficient Conditions for Skew Generalized Power Series Module M[[S,w]] to be T[[S,w]]-Noetherian R[[S,w]]-module,” Al-Jabar J. Pendidik. Mat., vol. 10, no. 2, pp. 285–292, 2019.
[17] T. Y. Lam, A First Course in Noncommutative Rings. New York: Springer-Verlag, 1991.
[18] S. Rugayah, A. Faisol, and Fitriani, “Matriks atas Ring Deret Pangkat Tergeneralisasi Miring,” BAREKENG J. Ilmu Mat. dan Terap., vol. 15, no. 1, pp. 157– 166, 2021.
[19] A. Kovacs, “Homomorphisms of Matrix Rings into Matrix Rings,” Pacific J. Math., vol. 49, no. 1, pp. 161–170, 1973.
[20] Y. Wang and Y. Wang, “Jordan homomorphisms of upper triangular matrix rings,” Linear Algebra Appl., vol. 439, no. 12, pp. 4063–4069, 2013.
[21] Y. Du and Y. Wang, “Jordan homomorphisms of upper triangular matrix rings over a prime ring,” Linear Algebra Appl., vol. 458, pp. 197–206, 2014.
[22] P. Ribenboim, “Rings of Generalized Power Series: Nilpotent Elements,” Abh. Math. Sem. Univ. Hambg., vol. 61, pp. 15–33, 1991.
[23] G. A. Elliott and P. Ribenboim, “Fields of generalized power series,” Arch. der Math., vol. 54, no. 4, pp. 365–371, 1990.
[24] M. Ziembowski, “Right Gaussian Rings and Related Topics,” University of Edinburgh, 2010.