We construct a nil ring R which has bounded index of nilpotence 2, is Wedderburn radical, and is commutative, and which also has a derivation δ for which the differential polynomial ring R[x; δ] is not even prime radical. This example gives a strong barrier to lifting certain radical properties from rings to differential polynomial rings. It also demarcates the strength of recent results about ...

A commutative ring A is said to be clean if every element of A can be written as a sum of a unit and an idempotent. This definition dates back to 1977 where it was introduced by W. K. Nicholson [7]. In 2002, V. P. Camillo and D. D. Anderson [1] investigated commutative clean rings and obtained several important results. In [4] Han and Nicholson show that if A is a semiperfect ring, then A[Z2] i...

We completely determine those natural numbers $n$ for which the full matrix ring $M_n(F_2)$ and triangular $T_n(F_2)$ over two elements field $F_2$ are either n-torsion clean or almost clean, respectively. These results somewhat address settle a question, recently posed by Danchev-Matczuk in Contemp. Math. (2019) as well they supply more precise aspect nil-cleanness property of $n\times n$ all ...