Rings in which every element is either a sum or a difference of a nilpotent and an idempotent (Q2816917)

An element in a unital ring is called \textit{weakly nil-clean} if it is either a sum or a difference of a nilpotent and an idempotent, and a ring is \textit{weakly nil-clean} is all its elements are so.NEWLINENEWLINEA selection of results is the following.NEWLINENEWLINETheorem 2. Let \(R\) be a ring. The following conditions are equivalent:NEWLINENEWLINE(1) \(R\) is weakly nil-clean;NEWLINENEWLINE(2) \(6\) is nilpotent and \(R/6R\) is weakly nil-clean;NEWLINENEWLINE(3) \(R/J(R)\) is weakly nil-clean and \(J(R)\) is nil.NEWLINENEWLINEProposition 6. A ring \(R\) is nil-clean iff \(R\) is weakly nil-clean and \(2\in J(R)\).NEWLINENEWLINETheorem 7. A ring \(R\) is weakly nil-clean iff \( R\cong R_{1}\times R_{2}\) where \(R_{1}\) is nil-clean and \( R_{2}\) is \(0\) or an indecomposable weakly nil-clean ring with \(3\in J(R_{2})\).NEWLINENEWLINECorollary 8. Weakly nil-clean rings are clean.NEWLINENEWLINEIn Theorem 12, the structure of abelian weakly nil-clean rings is determined. In particular, reduced weakly nil-clean rings are commutative.NEWLINENEWLINETheorem 12. Let \(D\) be a division ring and \(n\geq 1\). Then the matrix ring of \(n\times n\) matrices over \(D\) is weakly nil-clean iff \(D\cong \mathbb Z_{2}\), or, \(D\cong \mathbb Z_{3}\) and \(n=1\).