On strongly nil clean rings

A ring R is strongly nil clean if every element in R is the sum of an idempotent and a nilpotent that commutes. We prove, in this article, that a ring R is strongly nil clean if and only if for any a ∈ R, there exists an idempotent e ∈ ℤ[a] such that a − e ∈ N(R), if and only if R is periodic and R∕J(R) is Boolean, if and only if each prime factor ring of R is strongly nil clean. Further, we prove that R is strongly nil clean if and only if for all a ∈ R, there exist n ∈ ℕ,k ≥ 0 (depending on a) such that an−an+2k∈N(R), if and only if for fixed m,n ∈ ℕ, a−a2n+2m(n+1)∈N(R) for all a ∈ R. These also extend known theorems, e.g, [5, Theorem 3.21], [6, Theorem 3], [7, Theorem 2.7] and [12, Theorem 2].