Nonnil-S-Laskerian Rings

TL;DR

This paper introduces nonnil-S-Laskerian rings, a new class generalizing existing rings, and explores their properties, including their relation to nonnil-S-Noetherian rings and conditions under which their spectra are S-Noetherian.

Contribution

It defines nonnil-S-Laskerian rings, establishes their relationship with nonnil-S-Noetherian rings, and investigates their spectral properties and behavior in power series rings.

Findings

Nonnil-S-Noetherian rings are a subclass of nonnil-S-Laskerian rings.

Nonnil-S-Laskerian rings have S-Noetherian spectra under mild conditions.

Power series rings over nonnil-S-Laskerian rings with S-decomposable nilradical imply the base ring is S-laskerian.

Abstract

In this paper, we introduce the concept of nonnil-S-Laskerian rings, which generalize both nonnil-Laskerian rings and S-Laskerian rings. A ring R is said to be nonnil-S-Laskerian if every nonnil ideal I (disjoint from S) of R is S-decomposable. As a main result, we prove that the class of nonnil-S-Noetherian rings belongs to the class of nonnil-S-Laskerian rings. Also, we prove that a nonnil-S-Laskerian ring has S-Noetherian spectrum under a mild condition. Among other results, we prove that if the power series ring R[[X]] is nonnil-S-Laskerian with S-decomposable nilradical, then R is S-laskerian and satisfies the S-SFT property.