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

Authors : Andrada Ciımpean, Peter Danchev

Pages : 180-197

Doi:10.24330/ieja.296326

View : 16 | Download : 12

Publication Date : 2017-01-17

Article Type : Research Paper

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 T2insert ignore into journalissuearticles values(Zp);, T3insert ignore into journalissuearticles values(Zp); and M2insert ignore into journalissuearticles values(Z3);, respectively, as well as we establish a result on the weakly nilclean index of Mninsert ignore into journalissuearticles values(R); whenever R is a ring. Our results considerably extend and correct the corresponding ones from [Int. Electron. J. Algebra 15insert ignore into journalissuearticles values(2014);, 145–156]
Keywords : Nil clean ring, uniquely nil clean ring, weakly nil clean ring