Rings with a finite set of nonnilpotents

Let R be a ring and let N denote the set of nilpotent elements of R . Let n be a nonnegative integer. The ring R is called a θ n -ring if the number of elements in R which are not in N is at most n . The following theorem is proved: If R is a θ n -ring, then R is nil or R is finite. Conversely, if R is a nil ring or a finite ring, then R is a θ n -ring for some n . The proof of this theorem uses the structure theory of rings, beginning with the division ring case, followed by the primitive ring case, and then the semisimple ring case. Finally, the general case is considered.