WEAKLY NIL-CLEAN INDEX AND UNIQUELY WEAKLY NIL-CLEAN RINGS

Abstract

  We introduce and study the weakly nil-clean index associated to a ring. We also give some simple properties of this index and show that rings with the weakly nil-clean index 1 are precisely those rings that are abelian weakly nil-clean, thus showing that they coincide with uniquely weakly nil-clean rings. Next, we define certain types of nilpotent elements and weakly nil-clean decompositions by obtaining some results when the weakly nil-clean index is at most 2 and, moreover, we somewhat characterize rings with weakly nil-clean index 2. After that, we compute the weakly nil-clean index for T2(Zp), T3(Zp) and M2(Z3), respectively, as well as we establish a result on the weakly nilclean index of Mn(R) whenever R is a ring. Our results considerably extend and correct the corresponding ones from [Int. Electron. J. Algebra 15(2014), 145–156]

Keywords

• : Nil-clean ring,uniquely nil-clean ring,weakly nil-clean ring

References

1. [1] D. K. Basnet and J. Bhattacharyya, Nil clean index of rings, Int. Electron. J.
2. Algebra, 15 (2014), 145–156.
3. [2] S. Breaz, G. C˘alug˘areanu, P. Danchev and T. Micu, Nil-clean matrix rings,
4. Linear Algebra Appl., 439(10) (2013), 3115–3119.
5. [3] S. Breaz, P. Danchev and Y. Zhou, Rings in which every element is either a
6. sum or a difference of a nilpotent and an idempotent, J. Algebra Appl., 15(8) (2016), 1650148, 11 pp.
7. [4] S. Breaz and G. C. Modoi, Nil-clean companion matrices, Linear Algebra Appl., 489 (2016), 50–60.
8. [5] H. Chen, On uniquely clean rings, Comm. Algebra, 39(1) (2011), 189–198

9. [6] P. V. Danchev, Weakly UU rings, Tsukuba J. Math., 40(1) (2016), 101–118.
10. [7] P. V. Danchev and W. Wm. McGovern, Commutative weakly nil clean unital rings, J. Algebra, 425 (2015), 410-422.
11. [8] A. J. Diesl, Nil clean rings, J. Algebra, 383 (2013), 197–211.
12. [9] T. Y. Lam, A First Course in Noncommutative Rings, Second Edition, Grad.
13. Texts in Math., 131, Springer-Verlag, New York, 2001.
14. [10] T. K. Lee and Y. Zhou, Clean index of rings, Comm. Algebra, 40(3) (2012), 807–822.
15. [11] T. K. Lee and Y. Zhou, Rings of clean index 4 and applications, Comm. Algebra,
16. 41(1) (2013), 238–259.
17. [12] J. Ster, Nil clean involutions, arXiv preprint, (2015).