Cohen, Levitzki, Hilbert Basis, and Lasker-Noether Theorems for Nil-S-Noetherian Rings

In this paper, we introduce a new class of rings called Nil-$S$-Noetherian rings, which generalizes both $S$-Noetherian rings and $Ni{l}_{\ast }$-Noetherian rings. We investigate several properties of this new class and establish generalized versions of some classical results, including Cohen’s theorem, Levitzki’s theorem, and Hilbert’s basis theorem. Furthermore, we prove $S$-version of classical Lasker-Noether theorem for Nil-$S$-Noetherian rings.