The Ring Homomorphisms of Matrix Rings over Skew Generalized Power Series Rings

Ahmad Faisol, Fitriani Fitriani

Abstract


Let  M_n (R_1 [[S_1,≤_1,ω_1]]) and M_n (R_2 [[S_2,≤_2,ω_2]]) be a matrix rings over skew generalized power series rings, where R_1,R_2 are commutative rings with an identity element, (S_1,≤_1 ),(S_2,≤_2 ) are strictly ordered monoids, ω_1:S_1→End(R_1 ),〖 ω〗_2:S_2→End(R_2 ) are monoid homomorphisms. In this research, a mapping  τ from M_n (R_1 [[S_1,≤_1,ω_1]]) to M_n (R_2 [[S_2,≤_2,ω_2]]) is defined by using a strictly ordered monoid homomorphism δ:(S_1,≤_1 )→(S_2,≤_2 ), and ring homomorphisms μ:R_1→R_2 and σ:R_1 [[S_1,≤_1,ω_1]]→R_2 [[S_2,≤_2,ω_2]]. Furthermore, it is proved that τ is a ring homomorphism, and also the sufficient conditions for  τ to be a monomorphism, epimorphism, and isomorphism are given.

Keywords


matrix rings; homomorphisms; strictly ordered monoid; skew generalized power series rings.

Full Text:

PDF

References


H. Anton and C. Rorres, Elementary Linear Algebra: Applications Version, 9th Edition. New Jersey, 2005.

W. C. Brown, Matrices Over Commutative Rings. New York: Marcel Dekker Inc., 1993.

D. S. Dummit and R. M. Foote, Abstract Algebra, Third Edit. John Wiley and Sons, Inc., 2004.

R. Mazurek and M. Ziembowski, “Uniserial rings of skew generalized power series,” J. Algebr., vol. 318, no. 2, pp. 737–764, 2007.

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.

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.

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.

R. Mazurek, “Rota-Baxter operators on skew generalized power series rings,” J. Algebr. its Appl., vol. 13, no. 7, pp. 1–10, 2014.

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.

R. Mazurek and K. Paykan, “Simplicity of skew generalized power series rings,” New York J. Math., vol. 23, pp. 1273–1293, 2017.

A. Faisol, “Homomorfisam Ring Deret Pangkat Teritlak Miring,” J. Sains MIPA, vol. 15, no. 2, pp. 119–124, 2009.

A. Faisol, “Pembentukan Ring Faktor Pada Ring Deret Pangkat Teritlak Miring,” in Prosiding Semirata FMIPA Univerisitas Lampung, 2013, pp. 1–5.

A. Faisol, “Endomorfisma Rigid dan Compatible pada Ring Deret Pangkat Tergeneralisasi Miring,” J. Mat., vol. 17, no. 2, pp. 45–49, 2014.

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.

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.

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.

T. Y. Lam, A First Course in Noncommutative Rings. New York: Springer-Verlag, 1991.

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.

A. Kovacs, “Homomorphisms of Matrix Rings into Matrix Rings,” Pacific J. Math., vol. 49, no. 1, pp. 161–170, 1973.

Y. Wang and Y. Wang, “Jordan homomorphisms of upper triangular matrix rings,” Linear Algebra Appl., vol. 439, no. 12, pp. 4063–4069, 2013.

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.

P. Ribenboim, “Rings of Generalized Power Series: Nilpotent Elements,” Abh. Math. Sem. Univ. Hambg., vol. 61, pp. 15–33, 1991.

G. A. Elliott and P. Ribenboim, “Fields of generalized power series,” Arch. der Math., vol. 54, no. 4, pp. 365–371, 1990.

M. Ziembowski, “Right Gaussian Rings and Related Topics,” University of Edinburgh, 2010.




DOI: https://doi.org/10.18860/ca.v7i1.13001

Refbacks

  • There are currently no refbacks.


Copyright (c) 2021 Ahmad Faisol, Fitriani Fitriani

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Editorial Office
Mathematics Department,
Universitas Islam Negeri Maulana Malik Ibrahim Malang
Gajayana Street 50 Malang, East Java, Indonesia 65144
Faximile (+62) 341 558933
e-mail: cauchy@uin-malang.ac.id

Creative Commons License
CAUCHY: Jurnal Matematika Murni dan Aplikasi is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.