DOI: https://doi.org/10.14710/jfma.v5i1.11644

IDEMPOTENT MATRIX OVER SKEW GENERALIZED POWER SERIES RINGS

*Ahmad Faisol orcid  -  Universitas Lampung, Indonesia
Fitriani Fitriani orcid  -  Universitas Lampung, Indonesia

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Abstract

Let $R[[S,\leq,\omega]]$ be a skew generalized power series ring, with $R$ is a ring with an identity element, $(S,\leq)$ a strictly ordered monoid, and $\omega:S\rightarrow End(R)$ a monoid homomorphism. We define  the set of all matrices over $R[[S,\leq,\omega]]$, denoted by $M_{n}(R[[S,\leq,\omega]])$. With the addition and multiplication matrix operations, $M_{n}(R[[S,\leq,\omega]])$ becomes a ring. In this paper, we determine the sufficient conditions for $R$, $(S,\leq)$, and $\omega$, so the element of $M_{n}(R[[S,\leq,\omega]])$ is an idempotent matrix. 

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Keywords: idempotent matrix; matrices over a ring; strictly ordered monoid; monoid homomorphism; skew generalized power series ring
Funding: Research Institutions and Community Service of Universitas Lampung under contract No: 1648/UN 26.21/PN/2021

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Section: FUNDAMENTAL MATHEMATICS AND APPLICATIONS
Language : EN
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  1. H. Anton and C. Rorres, {em Elementary Linear Algebra: Applications Version, 9th Edition}. New Jersey, 2005
  2. W. C. Brown, {em Matrices Over Commutative Rings}. New York: : Marcel Dekker Inc., 1993
  3. D. S. Dummit and R. M. Foote, {em Abstract Algebra, Third Edit}. John Wiley and Sons, Inc., 2004
  4. R. Mazurek and M. Ziembowski, “Uniserial Rings of Skew Generalized Power Series,” {em J. Algebra}, vol. 318, no. 2, pp. 737-764, 2007
  5. R. Mazurek and M. Ziembowski, “On Von Neumann Regular Rings of Skew Generalized Power Series,” {em Communications in Algebra}, 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,” {em 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,” {em J. Math. Soc. Japan}, vol. 62, no. 4, pp. 1093-1112, 2010
  8. A. Faisol, “Homomorfisam Ring Deret Pangkat Teritlak Miring,” J. Sains MIPA, vol. 15, no. 2, pp. 119-124, 2009
  9. A. Faisol, “Pembentukan Ring Faktor Pada Ring Deret Pangkat Teritlak Miring,” in Prosiding Semirata FMIPA Univerisitas Lampung, 2013, pp. 1-5
  10. A. Faisol, “Endomorfisma Rigid dan Compatible pada Ring Deret Pangkat Tergeneralisasi Miring,” J. Mat., vol. 17, no. 2, pp. 45-49, 2014
  11. 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
  12. 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
  13. 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
  14. P. Ribenboim, “Generalized Power Series Rings,” in Lattices, Semigroups, and Universal Algebra, 1990
  15. R. Gilmer, Commutative Semigroup Rings. Chicago and London: The University of Chicago, 1984
  16. T. W. Hungerford, Algebra. New York: Springer-Verlag New York, Inc., 1974
  17. P. Ribenboim, “Rings of Generalized Power Series: Nilpotent Elements,” Abh. Math. Sem. Univ. Hambg., vol. 61, pp. 15-33, 1991
  18. P. Ribenboim, “Noetherian Rings of Generalized Power Series,” J. Pure Appl. Algebr., vol. 79, no. 3, pp. 293-312, 1992
  19. P. Ribenboim, “Rings of Generalized Power Series II: Units and Zero-Divisors,” J. Algebr., vol. 168, pp. 71-89, 1994
  20. P. Ribenboim, “Special Properties of Generalized Power Series,” J. Algebr., vol. 173, pp. 566-586, 1995
  21. P. Ribenboim, “Semisimple Rings and Von Neumann Regular Rings of Generalized Power Series,” J. Algebr., vol. 198, no.2, pp. 327-338, 1997
  22. A. Faisol, B. Surodjo, and S. Wahyuni, “T[[S]]-Noetherian Property on Generalized Power Series Modules,” JP J. Algebr. Number Theory Appl., vol. 43, no. 1, pp. 1-12, 2019
  23. A. Faisol, B. Surodjo, and S. Wahyuni, “The Sufficient Conditions for R[X]-module M[X] to be S[X]-Noetherian,” European J. Math. Sci., vol. 5, no. 1, pp. 1-13, 2019
  24. A. Faisol, B. Surodjo, and S. Wahyuni, “The Relation between Almost Noetherian Module, Almost Finitely Generated Module and T-Noetherian Module,” in Journal of Physics: Conference Series, Sep. 2019, vol. 1306, no. 1
  25. W.A. Pardede, A. Faisol, and Fitriani, “The X[[S]]-Sub-Exact Sequence of Generalized Power Series Rings,” Al-Jabar: Jurnal Pendidikan Matematika, vol. 11, no. 2, pp. 299-306, 2020
  26. A. Faisol, Fitriani, and Sifriyani, “Determining the Noetherian Property of Generalized Power Series Modules by Using X-Sub-Exact Sequence,” in Journal of Physics: Conference Series, Jan. 2021, vol. 1751, no. 1
  27. T. Y. Lam, A First Course in Noncommutative Rings. Springer-Verlag, New York, Inc., 1991
  28. S. Rugayah, A. Faisol, and Fitriani, “Matriks atas Ring Deret Pangkat Tergeneralisasi Miring,” Barekeng: Jurnal Ilmu Matematika dan Terapan, vol. 15, no. 1, pp. 157-166, 2021
  29. W. A. Adkins and S. H. Weintraub, Algebra: An Approach via Module Theory. New York: Springer-Verlag, New York, Inc., 1992
  30. C. D. Meyer, Matrix Analysis and Applied Linear Algebra. SIAM, 2000

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