Nil-clean and strongly nil-clean rings》.pdfVIP

Journal of Pure and Applied Algebra 220 (2016) 633–646 Contents lists available at ScienceDirect Journal of Pure and Applied Algebra /locate/jpaa Nil-clean and strongly nil-clean rings Tamer Koşan a , Zhou Wang b , Yiqiang Zhou c,∗ a Department of Mathematics, Gebze Technical University, Gebze/Kocaeli, Turkey b Department of Mathematics, Southeast University, Nanjing 210096, PR China c Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Nfld A1C 5S7, Canada a r t i c l e i n f o a b s t r a c t Article history: An element a of a ring R is nil-clean if a = e + b where e2 = e ∈ R and b is a Received 28 August 2014 nilpotent; if further eb = be, the element a is called strongly nil-clean. The ring R Received in revised form 8 May 2015 is called nil-clean (resp., strongly nil-clean) if each of its elements is nil-clean (resp., Available online 5 August 2015 strongly nil-clean). It is proved that an element a is strongly nil-clean iff a is a sum Communicated by S. Iyengar 2 of an idempotent and a unit that commute and a −a is a nilpotent, and that a ring MSC: R is strongly nil-clean iff R/J (R) is boolean and J (R) is nil, where J (R) denotes the Primary: 16U99; secondary: 16E50; Jacobson radical of R. The strong nil-cleanness of Morita contexts, formal matrix 16N40; 16S34; 16S50; 16S70