n-fine rings

Abstract

. A ring R is said to be n -fine if every nonzero element in R can be written as a sum of a nilpotent and n units in R. The class of these rings contains fine rings and n -good rings in which each element is a sum of n units. Fundamental properties of such rings are obtained. One of the main results of this paper is that the m × m matrix ring M m ( R ) over any arbitrary ring R is 2-fine. Furthermore, the m × m matrix ring M m ( R ) over a n -fine ring R is n -fine.