Abstract: Let R be a commutative ring with identity element(1≠0), the set of nilpotent elements in ring R is an ideal of R, which is recorded as the nilradical of ring R, denoted by Nil(R). The ideals which are not included in the nilradicals in R are called non-nil ideals of R. Introduced in the paper are a few types of communicative rings with some nilradicals as the prime ideals like ZN-ring, NP-ring,-ring, strongly -ring and the connection and distinction of them are clarified with some examples. Some important properties of non-nil ideals are proved.