SUFFICIENT CONDITION FOR A NIL-CLEAN ELEMENT TO BE CLEAN IN A CERTAIN SUBRING OF M3(Z)

  • Muhamad Faikar Mustafidz Al Habibi
  • Santi Irawati
  • Hery Susanto
  • I Made Sulandra
  • Hidetoshi Marubayashi
  • Ida Fitriana Ambarsari
Keywords: nil-clean ring, clean ring, nil-clean element, clean element, matrices

Abstract

Diesl has proved that a nil-clean ring is clean. However, not all nil-clean element of any ring is clean as showed by Andrica by providing counter examples in 2 x 2 matrices over Z. The objective of this study is to determine sufficient condition for a nil-clean element tobe clean in a certain subring of M3(Z) . The two main methods are constructing certain subring, namely X3(Z) & M3(Z) of  and then identifying idempotent and nilpotent elements in X3(Z). This construction provides examples as the extension of those matrices founded by Andrica in the sense of matrix order and the different form of those matrices. The methods are used in finding the sufficient condition for nil-clean elements to be clean in a certain subring of M3(Z). By this finding, we follow up the previous researches especially from Diesl and Andrica. As the application, it is provided nil-clean elements in X3(Z) which are clean and some other elements which are not clean.

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References

W. K. Nicholson, "Lifting Idempotents And Exchange Rings",Trans. Amer. Math. Soc. 229pp 269–278, 1977

D. Andrica and G. Calugareanu, “A nil-clean 2 × 2 matrix over the integers which is not clean,†J. Algebra Appl.13,No 6 1450009, 9 pp, 2014.

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T. Andreescu, D. Andrica, and I. Cucurezeanu, Introduction to Diophantine Equations. Springer, 2010.

N. A. Immormino, R. J. Blok, and M. D. Staic, “Clean Rings & Clean Group Rings".Ph.D. dissertation, BowlingGreen State University

H. Rosen, J. G. Michaels, and J. W. Grossman, Handbook of Discrete and Combinatorial Mathematics. Florida: CRC Press LLC, 2000.

W. C. Brown, Matrices over Commutative Rings. Marcel Dekker, 1993.

Published
2019-10-03
How to Cite
Al Habibi, M. F. M., Irawati, S., Susanto, H., Sulandra, I. M., Marubayashi, H., & Ambarsari, I. F. (2019). SUFFICIENT CONDITION FOR A NIL-CLEAN ELEMENT TO BE CLEAN IN A CERTAIN SUBRING OF M3(Z). COMPUSOFT: An International Journal of Advanced Computer Technology, 8(9). Retrieved from https://www.ijact.in/index.php/ijact/article/view/1024