# Rings with uniformly S-w-Noetherian spectrum

## TL;DR

This paper introduces and characterizes rings with uniformly S-w-Noetherian spectrum, providing several equivalent conditions and extending classical results to this new class of rings.

## Contribution

It introduces the concept of rings with uniformly S-w-Noetherian spectrum and establishes multiple equivalent characterizations, including extensions to polynomial and localization rings.

## Key findings

- A ring has uniformly S-w-Noetherian spectrum iff each radical ideal is radically S-w-finite.
- Such rings have stationary ascending chains of radical w-ideals.
- The classical case is extended to include countably generated ideals and polynomial rings.

## Abstract

In this paper, the notion of rings with uniformly S-w-Noetherian spectrum is introduced. Several characterizations of rings with uniformly S-w-Noetherian spectrum are given. Actually, we show that a ring R has uniformly S-w-Noetherian spectrum with respect to some s 2 S if and only if each ascending chain of radical w-ideals of R is stationary with respect to s 2 S, if and only if each radical (prime) (w-)ideal of R is radically S-w-finite with respect to s, if and only if each countably generated ideal of R is radically S-w-finite with respect to s, if and only if R[X] has uniformly S-w-Noetherian spectrum, if and only if RfXg has uniformly S-Noetherian spectrum. Beside, we show a ring R has Noetherian (resp., uniformly S-Noetherian) spectrum with respect to s if and only if each countably generated ideal of R is radically finite (S-finite with respect to s), which is a new result in the classical case.