Group graded rings with the nil-good property

Decompositions of matrices into diagonalizable and square-zero matrices

“…On the same vein, a slightly more general class of rings than the aforementioned class of fine rings was defined in [5] under the name nil-good rings and some their characteristic properties, including the behaviour of the matrix ring over a nil-good ring, were explored in [4] and [7], respectively.…”

Decompositions of matrices into a sum of invertible matrices and matrices of fixed nilpotence

“…M 9 (F)e(1,4,4) where e (1,4,4) = e 1,1 + e 4,4 + e 5,5 + e 6,6 . Moreover, since r = 5 and k = 4, we get c = 1 and d = 0 in the formula r − k + 2 = c(k − 1) + d, so we also consider the matrix N…”

“…M 9 (F)e(2,7,4) where e (2,7,4) = e 2,2 + e 7,7 + e 8,8 + e 9,9 . Thus, N C = N 1,4,4 + N 2,7,4 satisfies N 4 C = 0 andC + N C =…”