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ON A GENERALIZATION OF UNIT REGULAR RINGS

  • Tahire Ozen (Department of Mathematics Bolu Abant Izzet Baysal University)
  • Received : 2022.08.18
  • Accepted : 2023.09.01
  • Published : 2023.11.30
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Abstract

In this paper, we introduce a class of rings generalizing unit regular rings and being a subclass of semipotent rings, which is called idempotent unit regular. We call a ring R an idempotent unit regular ring if for all r ∈ R - J(R), there exist a non-zero idempotent e and a unit element u in R such that er = eu, where this condition is left and right symmetric. Thus, we have also that there exist a non-zero idempotent e and a unit u such that ere = eue for all r ∈ R - J(R). Various basic characterizations and properties of this class of rings are proved and it is given the relationships between this class of rings and some well-known classes of rings such as semiperfect, clean, exchange and semipotent. Moreover, we obtain some results about when the endomorphism ring of a module in a class of left R-modules X is idempotent unit regular.

Keywords

Acknowledgement

I would like to thank the referee for his (or her) very important contribution to the development of this article.

References

  1. R. M. S. Alfaqi, On a special case of clean rings, Thesis (Master), Bolu Abant Izzet Baysal University, 2021. https://tez.yok.gov.tr/UlusalTezMerkezi/giris.jsp
  2. M. H. Ali and J. M. Zelmanowitz, Discrete implies continuous, J. Algebra 183 (1996), no. 1, 186-192. https://doi.org/10.1006/jabr.1996.0213
  3. V. P. Camillo and D. Khurana, A characterization of unit regular rings, Comm. Algebra 29 (2001), no. 5, 2293-2295. https://doi.org/10.1081/AGB-100002185
  4. V. P. Camillo, D. Khurana, T. Y. Lam, W. K. Nicholson, and Y. Zhou, A short proof that continuous modules are clean, in Contemporary ring theory 2011, 165-169, World Sci. Publ., Hackensack, NJ, 2012. https://doi.org/10.1142/9789814397681_0015
  5. H. Chen, Extensions of unit-regular rings, Comm. Algebra 32 (2004), no. 6, 2359-2364. https://doi.org/10.1081/AGB-120037225
  6. H. Chen, On strongly J-clean rings, Comm. Algebra 38 (2010), no. 10, 3790-3804. https://doi.org/10.1080/00927870903286835
  7. H. Chen, On uniquely clean rings, Comm. Algebra 39 (2011), no. 1, 189-198. https://doi.org/10.1080/00927870903451959
  8. A. J. Diesl, Nil clean rings, J. Algebra 383 (2013), 197-211. https://doi.org/10.1016/j.jalgebra.2013.02.020
  9. G. Ehrlich, Unit-regular rings, Portugal. Math. 27 (1968), 209-212.
  10. K. R. Goodearl, von Neumann Regular Rings, second edition, Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1991.
  11. N. A. Immormino, Clean rings & clean group rings, Thesis (Ph.D.), Bowling Green State University, 2013.
  12. T.-M. Lok and K. P. Shum, On matrix rings over unit-regular rings, Beitrage Algebra Geom. 44 (2003), no. 1, 179-188.
  13. S. H. Mohamed and B. J. Muller, Continuous and Discrete Modules, London Mathematical Society Lecture Note Series, 147, Cambridge Univ. Press, Cambridge, 1990. https://doi.org/10.1017/CBO9780511600692
  14. W. K. Nicholson, I-rings, Trans. Amer. Math. Soc. 207 (1975), 361-373. https://doi.org/10.2307/1997182
  15. W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269-278. https://doi.org/10.2307/1998510
  16. W. K. Nicholson, K. Varadarajan, and Y. Zhou, Clean endomorphism rings, Arch. Math. (Basel) 83 (2004), no. 4, 340-343. https://doi.org/10.1007/s00013-003-4787-9
  17. M. O Searcoid, Perturbation of linear operators by idempotents, Irish Math. Soc. Bull. No. 39 (1997), 10-13.
  18. J. Ster, Corner rings of a clean ring need not be clean, Comm. Algebra 40 (2012), no. 5, 1595-1604. https://doi.org/10.1080/00927872.2011.551901
  19. A. A. Tuganbaev, Rings Close to Regular, Mathematics and its Applications, 545, Kluwer Acad. Publ., Dordrecht, 2002. https://doi.org/10.1007/978-94-015-9878-1