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On $S$-$n$-absorbing ideals

Hyungtae Baek, Hyun Seung Choi, Jung Wook Lim

TL;DR

This work introduces $S$-$n$-absorbing ideals as a unifying generalization of $S$-prime and $n$-absorbing ideals in commutative algebra. It develops a broad framework: (i) foundational properties and counterexamples, (ii) quotient-extension behavior and precise equivalence conditions under $S$-Laskerian and locally divided hypotheses, (iii) the minimality function $\omega_{R,S}(I)$ and the spectrum $\Omega_S(R)$, including their behavior under homomorphisms, products, and amalgamations, and (iv) the behavior of $S$-$n$-absorbing ideals under amalgamation with explicit constructions and limitations. The results connect generalized Noetherianity notions with multiplicative ideal theory, offering tools to analyze $S$-$n$-absorbing ideals in extensions and amalgamations and to quantify their complexity via $\omega_{R,S}(I)$ and $\Omega_S(R)$.

Abstract

Let $R$ be a commutative ring with identity, $S$ a multiplicative subset of $R$ and $I$ an ideal of $R$ disjoint from $S$. In this paper, we introduce the notion of an $S$-$n$-absorbing ideal which is a generalization of both the $S$-prime ideals and $n$-absorbing ideals. Moreover, we investigate the basic properties, quotient extension, existence and amalgamation of $S$-$n$-absorbing ideals.

On $S$-$n$-absorbing ideals

TL;DR

This work introduces --absorbing ideals as a unifying generalization of -prime and -absorbing ideals in commutative algebra. It develops a broad framework: (i) foundational properties and counterexamples, (ii) quotient-extension behavior and precise equivalence conditions under -Laskerian and locally divided hypotheses, (iii) the minimality function and the spectrum , including their behavior under homomorphisms, products, and amalgamations, and (iv) the behavior of --absorbing ideals under amalgamation with explicit constructions and limitations. The results connect generalized Noetherianity notions with multiplicative ideal theory, offering tools to analyze --absorbing ideals in extensions and amalgamations and to quantify their complexity via and .

Abstract

Let be a commutative ring with identity, a multiplicative subset of and an ideal of disjoint from . In this paper, we introduce the notion of an --absorbing ideal which is a generalization of both the -prime ideals and -absorbing ideals. Moreover, we investigate the basic properties, quotient extension, existence and amalgamation of --absorbing ideals.
Paper Structure (5 sections, 31 theorems)

This paper contains 5 sections, 31 theorems.

Table of Contents

  1. Introduction
  2. Basic properties of $S$-$n$-absorbing ideals
  3. $S$-$n$-absorbing ideals of quotient rings
  4. The functions $\omega_{R,S}(I)$ and $\Omega_S(R)$
  5. $S$-$n$-absorbing ideals of amalgamation

Key Result

Proposition 2.3

Let $R$ be a commutative ring with identity and let $S,S_1,\dots,S_m$ be multiplicative subsets of $R$. Then the following assertions hold.

Theorems & Definitions (63)

  • Definition 2.1
  • Example 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • proof
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • Corollary 2.7
  • ...and 53 more