On nonnil-S-Noetherian and nonnil-u-S-Noetherian rings

Let R be a commutative ring with identity, and let S be a multiplicative subset of R. Then R is called a nonnil-S-Noetherian ring if every nonnil ideal of R is S-finite. Also, R is called a u-S-Noetherian ring if there exists an element s ∈ S such that for each ideal I of R, sI ⊆ K for some finitely generated sub-ideal K of I. In this paper, we examine some new characterization of nonnil-S-Noetherian rings. Then, as a generalization of nonnil-S-Noetherian rings and u-S-Noetherian rings, we introduce and investigate the nonnilu-S-Noetherian rings class.