2015/10/26 by Basnet, Dhiren Kumar, Bhattacharyya, Jayanta
#16N40 #16U99 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1510.07440
We introduce the concept of a weak nil clean ring, a generalization of nil clean ring, which is nothing but a ring with unity in which every element can be expressed as sum or difference of a nilpotent and an idempotent. Further if the idempotent and nilpotent commute the ring is called weak* nil clean. We characterize all n∈ ℕ, for which ℤn is weak nil clean but not nil clean. We show that if R is a weak* nil clean and e is an idempotent in R, then the corner ring eRe is also weak* nil clean. Also we discuss S-weak nil clean rings and their properties, where S is a set of idempotents and show that if S=\0, 1\, then a S-weak nil clean ring contains a unique maximal ideal. Finally we show that weak* nil clean rings are exchange rings and strongly nil clean rings provided 2∈ R is nilpotent in the later case. We have ended the paper with introduction of weak J-clean rings.