<p>Nonsingular rings are a generalization of reduced rings, and therefore, nilpotent elements play an important role in this class of rings. There are many examples of rings with nilpotent elements which are nonsingular. To find a noncommutative proper subclass of nonsingular rings, we introduce the concepts of <i>N</i>-nonsingular rings. We show that for commutative rings, the concepts of <i>N</i>-nonsingular, nonsingular and reduced rings are the same. Also, we introduce <i>N</i>-singular rings as a generalization of singular rings. Moreover, we give some examples of these rings and consider the closure of the <i>N</i>-nonsingular and <i>N</i>-singular rings with respect to various extensions including different types of matrices rings, polynomial rings, etc.</p>