## Classification of Rings Satisfying Some Constraints on Subsets

- Moharram A. Khan, M. A. KHAN
- 2007

Let R be a ring (not necessarily with identity), N the set of nilpotents, and n > a fixed integer. Suppose that (i) N is commutative; (ii) If x N and n n y N, then x y xy (iii) For a N and b R, if n![a,b] 0, then [a,b] 0, where [a,b] ab ba denotes the commutator. Then R is commutative. This theorem generalizes the "x --x" theorem of Jacobson. It is also… (More)