Throughout this paper, all rings are associative rings with identity. The prime radical of a ring R and the set of nilpotent elements in R are denoted by P(R) and N(R), respectively. In this paper we show that a

ring R, in which N(R) forms a 2-sided ideal, is an (S,2)-ring if and only if every idempotent in R is a sum of two units in R, this result may be a generalization of [1, Theorem 4] on such rings and moreover on 2-primal rings. For a commutative

ring R, if 2 = 1 + 1 is a unit (i.e. invertible element) in R then R is an (S,2)-ring. As generalizations of commutative rings, there are PI-rings, 2-primal rings and duo rings etc. By Fisher-Snider [5], the preceding argument is also true for PI-rings. As another generalization, Badawi [1] proves that for duo rings the result holds. In this paper we obtain the result, as a corollary of our main result, on a ring R in which N(R) forms a 2-sided ideal (we call such a ring an NI-ring for simplicity in this paper), hiring the method of proof in [1] partially. Then we also get the result on 2-primal rings because 2-primal rings are NI-rings. The term 2-primal was come upon originally by Birkenmeir, Heatherly and Lee [2] in the context of left near rings. Y. Hirano, using the term N-ring for what is called a 2-primal ring, considered the 2-primal condition in the context of strongly

rings. He showed in [7] that for an N-ring R, R is strongly

if and only if

is strongly

for n = 1,2,.... G. Shin [9] proved that a ring R is 2-primal if and only if every minimal prime ideal of R is completely prime, which was one of the earliest results known to us about 2-primal rings (although not so called at the time.) Moreover we study the connections between 2-primal rings and NI-rings under some conditions.