(WEAKLY)n NIL CLEANNESS OF THE RING Zm

Yıl 2018, Cilt: 67 Sayı: 2, 29 - 37, 01.08.2018

A.hani Khashan H.ali Handam

Öz

Let R be an associative ring with identity. For a positive integer n > 2, an element a 2 R is called n potent if a n = a . We define R to be (weakly) n-nil clean if every element in R can be written as a sum (a sum or a difference) of a nilpotent and an npotent
element in R. This concept is actually a generalization of weakly nil clean rings introduced by Danchev and McGovern, [6]. In this paper, we completely determine all n; m 2 N such that the ring of integers modulo m, Zm is (weakly) nnil
clean

Anahtar Kelimeler

Nil clean ring, n-nil clean ring, weakly n-nil clean ring

Kaynakça

• Badawi, A., Chin A. Y. M. and Chen, H. V., On rings with near idempotent elements, International J. of Pure and Applied Math 1 (3) (2002), 255-262.
• Breaz, S., Danchev P. and Zhou, Y., Rings in which every element is either a sum or a diğerence of a nilpotent and an idempotent, preprint arXiv:1412.5544 [math.RA].
• Chen, H., Strongly nil clean matrices over R[x]=(x2 , Bull. Korean Math. Soc, 49 (3)(2012), 599.
• Chen, H. and Sheibani, M., Strongly 2-nil-clean rings, J. Algebra Appl., 16 (2017) DOI: 1142/S021949881750178X.
• Chen, H., On Strongly Nil Clean Matrices, Comm. Algebra, 41 (3) (2013), 1074-1086.
• Danchev, P.V. and McGovern, W.Wm., Commutative weakly nil clean unital rings, J. Algebra (2015), 410–422.
• Diesl, A. J., Classes of Strongly Clean Rings, Ph.D. Thesis, University of California, Berkeley, Diesl, A. J., Nil clean rings, J. Algebra, 383 (2013), 197-211.
• Hirano, Y., Tominaga H. and Yaqub, A., On rings in which every element is uniquely ex- pressible as a sum of a nilpotent element and a certain potent element, Math. J. Okayama Univ. 30 (1988), 33-40.
• Khashan, H. A. and Handam, A. H., g(x) nil clean rings, Scientiae Mathematicae Japonicae, , (2) (2016), 145-154.
• Khashan, H. A. and Handam, A. H., On weakly g(x)-nil clean rings, International J. of Pure and Applied Math, 114 (2) (2017), 191-202.
• Handam, A. H. and Khashan, H. A., Rings in which elements are the sum of a nilpotent and a root of a …xed polynomial that commute, Open Mathematics, 15 (1), (2017), 420-426.
• Nagell, T., Introduction to Number Theory. New York: Wiley, p. 157, 1951.
• Nicholson, W. K., Lifting idempotents and exchange rings, Trans. Amer. Math. Soc., 229 (1977), 269–278.