DOI: https://doi.org/10.14710/jfma.v6i2.19307

IDEMPOTENT ELEMENTS IN MATRIX RING OF ORDER 2 OVER POLYNOMIAL RING $\mathbb{Z}_{p^2q}[x]$

*Muchammad Choerul Arifin orcid  -  Dept. of Mathematics, Gadjah Mada University, Indonesia
Iwan Ernanto  -  Dept. of Mathematics, Gadjah Mada University, Indonesia

Citation Format:
Abstract
An idempotent element in the algebraic structure of a ring is an element that, when multiplied by itself, yields an outcome that remains unchanged and identical to the original element. Any ring with a unity element generally has two idempotent elements, 0 and 1, these particular idempotent elements are commonly referred to as the trivial idempotent elements However, in the case of rings $\mathbb{Z}_n$ and $\mathbb{Z}_n[x]$ it is possible to have non-trivial idempotent elements. In this paper, we will investigate the idempotent elements in the polynomial ring $\mathbb{Z}_{p^2q}[x]$ with $p,q$ different primes. Furthermore, the form and characteristics of non-trivial idempotent elements in $M_2(\mathbb{Z}_{p^2q}[x])$ will be investigated. The results showed that there are 4 idempotent elements in $\mathbb{Z}_{p^2q}[x]$ and 7 idempotent elements in $M_2(\mathbb{Z}_{p^2q}[x])$.
Fulltext View|Download

Article Metrics:

Article Info
Section: FUNDAMENTAL MATHEMATICS AND APPLICATIONS
Language : EN
Recent articles
A BACKTRACKING APPROACH FOR SOLVING PATH PUZZLES MATHEMATICAL MODEL OF MEASLES DISEASE SPREAD WITH TWO-DOSE VACCINATION AND TREATMENT TOPOLOGY OF QUASI-PSEUDOMETRIC SPACES AND CONTINUOUS LINEAR OPERATOR ON ASYMMETRIC NORMED SPACES More recent articles
  1. Malik D. S., J. N. Mordeson and M. K. Son, Fundamentals of Abstract Algebra. International Edition. United States of America: The McGraw-Hill Publishing Company, 1997
  2. Herstein, I. N, Topics in Algebra. 2nd Edition, New York: John Wiley and Sons, 1975
  3. Malman, Bartosz, ”Zero-Divisors and Indempotents In Group Rings” [Master’s Thesis,
  4. Lund University], Centrum Scientarium Mathematicarum, 2014
  5. Thomas Q. Sibley, “Idempotents ́a La Mod”, The College Mathematics Journal, vol. 43,
  6. no. 5, pp. 401–404, 2012
  7. K. R. Goodearl, Von Neumann Regular Rings. Second edition, Malabar: Robert E. Krieger
  8. Publishing Co., Inc., 1991
  9. Nicholson, W. Keith, ”Lifting Idempotents and Exchange Rings”, Trans. Amer. Math
  10. Soc., vol. 229, pp. 269-278, 1977
  11. Nicholson, W. Keith, ”Strongly Clean Rings and Fitting’s Lemma”, Communications in
  12. Algebra. vol. 27(8), pp. 3583-3592, 1999
  13. Kanwar, P., A. Leroy, and J. Matczuk, ”Clean Elements in Polynomial Rings”, Noncom-
  14. mutative Rings and their Applications, Contemp. Math., Amer. Math. Soc. pp: 197-204,
  16. R. K. Sharma, Pooja Yadav, and P. Kanwar, ”Lie Regular Generators of General Linear
  17. Groups”, Communications in Algebra. vol 40, pp. 1304-1315, 2012
  18. Kanwar, P., A. Leroy and J. Matczuk, ”Idempotents in Ring Extensions”, Journal of Al-
  19. gebra, vol. 389, pp. 128-136, 2013
  20. Kanwar, P., M. Khatkar and R. K. Sharma, ”Idempotents and Units of Matrix Rings Over
  21. Polynomial Rings”, International Electronic Journal of Algebra, vol. 22, pp. 147-169, 2017
  22. Jose Maria, P. Balmaceda, and Joanne Pauline P. Datu, ”Idempotents In Certain Matrix
  23. Rings Over Polynomial Rings”, International Electronic Journal of Algebra, vol. 27, pp
  24. -12, 2020
  25. C. Ding, D. Pei, and A. Salomaa, Chinese Remainder Theorem: Applications in Comput-
  26. ing, Coding, Cryptography. Singapore: World Scientific Publishing Co. Pte. Ltd, 1996
  27. Koshy Thomas, Elementary Number Theory with Applications. 2nd Edition. United King-
  28. dom: Acedemic Press of Elsevier, 2007
  29. Burton, David M, Elementary Number Theory. 6th Edition. New Delhi: Tata McGraw-
  30. Hill Publishing Company Limited, 2007

Last update:

No citation recorded.

Last update:

No citation recorded.