JOURNAL OF ALGEBRA AND ITS APPLICATIONS, vol.19, no.4, 2020 (SCI-Expanded)
For n >= 2 and for a ring R, the notation P-n(R) means that a(n) - a is nilpotent for all a is an element of R, and Q(n)(R) means that R/J(R) has identity x(n) = x and J(R) is nil, where J(R) is the Jacobson radical of R. In this paper, rings R for which P-n(R) holds are completely characterized for integers n up to 12, for n = 2(k) with k odd, and for n = 2(k)p, where k > 0 and p = 3,5,7 or 9. For an arbitrary integer n >= 2, we prove that P-n(R) double left right arrow Q(n)(R) for all rings R if and only if n is even with n 1 (mod 3), or n is odd with n not equivalent to 1 (mod 3) and n not equivalent to 1 (mod 8).