Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON g(x)-INVO CLEAN RINGS

  • El Maalmi, Mourad (Faculty of Sciences Dhar El Mahraz Sidi Mohamed Ben Abdellah University) ;
  • Mouanis, Hakima (Faculty of Sciences Dhar El Mahraz Sidi Mohamed Ben Abdellah University)
  • Received : 2019.04.20
  • Accepted : 2019.11.15
  • Published : 2020.04.30
Download PDF
⟨ Previous Next ⟩

Abstract

An element in a ring R with identity is called invo-clean if it is the sum of an idempotent and an involution and R is called invoclean if every element of R is invo-clean. Let C(R) be the center of a ring R and g(x) be a fixed polynomial in C(R)[x]. We introduce the new notion of g(x)-invo clean. R is called g(x)-invo if every element in R is a sum of an involution and a root of g(x). In this paper, we investigate many properties and examples of g(x)-invo clean rings. Moreover, we characterize invo-clean as g(x)-invo clean rings where g(x) = (x-a)(x-b), a, b ∈ C(R) and b - a ∈ Inv(R). Finally, some classes of g(x)-invo clean rings are discussed.

Keywords

References

  1. M.-S. Ahn and D. D. Anderson, Weakly clean rings and almost clean rings, Rocky Mountain J. Math. 36 (2006), no. 3, 783-798. https://doi.org/10.1216/rmjm/1181069429
  2. M. M. Ali, Idealization and theorems of D. D. Anderson, Comm. Algebra 34 (2006), no. 12, 4479-4501. https://doi.org/10.1080/00927870600938837
  3. D. D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra 1 (2009), no. 1, 3-56. https://doi.org/10.1216/JCA-2009-1-1-3
  4. G. Borooah, A. J. Diesl, and T. J. Dorsey, Strongly clean matrix rings over commutative local rings, J. Pure Appl. Algebra 212 (2008), no. 1, 281-296. https://doi.org/10.1016/j.jpaa.2007.05.020
  5. V. Camillo and J. J. Simon, The Nicholson-Varadarajan theorem on clean linear transformations, Glasg. Math. J. 44 (2002), no. 3, 365-369. https://doi.org/10.1017/S0017089502030021
  6. V. P. Camillo and H.-P. Yu, Exchange rings, units and idempotents, Comm. Algebra 22 (1994), no. 12, 4737-4749. https://doi.org/10.1080/00927879408825098
  7. H. Chen, On uniquely clean rings, Comm. Algebra 39 (2011), no. 1, 189-198. https://doi.org/10.1080/00927870903451959
  8. M. Chhiti, N. Mahdou, and M. Tamekkante, Clean property in amalgamated algebras along an ideal, Hacet. J. Math. Stat. 44 (2015), no. 1, 41-49.
  9. P. V. Danchev, Invo-clean unital rings, Commun. Korean Math. Soc. 32 (2017), no. 1, 19-27. https://doi.org/10.4134/CKMS.c160054
  10. P. V. Danchev, Corners of invo-clean unital rings, Pure Mathematical Sciences (2018), 27-31. https://doi.org/10.12988/pms.2018.877
  11. M. D'Anna, C. A. Finocchiaro, and M. Fontana, Amalgamated algebras along an ideal, in Commutative algebra and its applications, 155-172, Walter de Gruyter, Berlin, 2009.
  12. M. D'Anna, C. A. Finocchiaro, and M. Fontana, Properties of chains of prime ideals in an amalgamated algebra along an ideal, J. Pure Appl. Algebra 214 (2010), no. 9, 1633-1641. https://doi.org/10.1016/j.jpaa.2009.12.008
  13. L. Fan and X. Yang, On rings whose elements are the sum of a unit and a root of a fixed polynomial, Comm. Algebra 36 (2008), no. 1, 269-278. https://doi.org/10.1080/00927870701665461
  14. J. Han and W. K. Nicholson, Extensions of clean rings, Comm. Algebra 29 (2001), no. 6, 2589-2595. https://doi.org/10.1081/AGB-100002409
  15. H. A. Khashan and A. H. Handam, g(x)-nil clean rings, Sci. Math. Jpn. 79 (2016), no. 2, 145-154.
  16. A. Lambert and T. G. Lucas, Nagata's principle of idealization in relation to module homomorphisms and conditional expectations, Kyungpook Math. J. 40 (2000), no. 2, 327-337.
  17. W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269-278. https://doi.org/10.2307/1998510
  18. W. K. Nicholson, Strongly clean rings and Fitting's lemma, Comm. Algebra 27 (1999), no. 8, 3583-3592. https://doi.org/10.1080/00927879908826649
  19. W. K. Nicholson and Y. Zhou, Endomorphisms that are the sum of a unit and a root of a fixed polynomial, Canad. Math. Bull. 49 (2006), no. 2, 265-269. https://doi.org/10.4153/CMB-2006-027-6
  20. Z. Wang and J. Chen, A note on clean rings, Algebra Colloq. 14 (2007), no. 3, 537-540. https://doi.org/10.1142/S1005386707000491
  21. G. Xiao and W. Tong, n-clean rings and weakly unit stable range rings, Comm. Algebra 33 (2005), no. 5, 1501-1517. https://doi.org/10.1081/AGB-200060531
  22. G. Xiao and W. Tong, n-clean rings, Algebra Colloq. 13 (2006), no. 4, 599-606. https://doi.org/10.1081/AGB-200060531
  23. Y. Ye, Semiclean rings, Comm. Algebra 31 (2003), no. 11, 5609-5625. https://doi.org/10.1081/AGB-120023977
  24. Z. Yi and Y. Zhou, Baer and quasi-Baer properties of group rings, J. Aust. Math. Soc. 83 (2007), no. 2, 285-296. https://doi.org/10.1017/S1446788700036909