On Z-Noetherian Rings

Abstract

In this paper, introduce a new finiteness condition for commutative rings, called $Z$-Noetherian, under which every ideal contained in the set of zero-divisors $Z(R)$ is finitely generated. This condition lies properly between Noetherian and $Nil_{*}$-Noetherian. We establish intrinsic characterizations of $Z$-Noetherian rings and investigate their stability under nilpotent quotients and localizations. Under the assumption that $Z(R)$ is an ideal, we obtain a primary decomposition theorem for ideals contained in $Z(R)$. For $ZD$-rings, we prove a Cohen-type criterion characterizing $Z$-Noetherian in terms of the finite generation of prime ideals contained in $Z(R)$. We further establish $Z$-Noetherian analogues of the Hilbert Basis Theorem and Krull's Intersection Theorem. Finally, we establish a principal ideal theorem for $Z$-Noetherian rings. Collectively, these results show that several fundamental finiteness and structural properties of Noetherian rings extend to the broader class of $Z$-Noetherian rings.