Ideal Containment in Commutative Rings

TL;DR

This paper explores the concepts of big and upper big ideals in commutative rings, providing new characterizations of Pr"ufer domains and analyzing these ideals in various ring constructions.

Contribution

It introduces and distinguishes big and upper big ideals, offering new characterizations of Pr"ufer domains and analyzing these ideals in pullback and trivial ring extension contexts.

Findings

Big and upper big ideals are fundamentally different.

Characterization of Pr"ufer domains via big ideal domains.

Identification of big ideals in rings with zero-divisors.

Abstract

Let $R$ be a commutative ring with identity. An ideal $I$ of $R$ is said to be a big ideal (resp. an upper big ideal) if whenever $J\subsetneqq I$ (resp. $I\subsetneqq J$), $J^{n}\subsetneqq I^{n}$ (resp. $I^{n}\subsetneqq J^{n}$) for every $n\geq 1$; and $R$ itself is a big ideal ring provided that every ideal of $R$ is a big ideal. In this paper we study the notions of big ideals, upper big ideals and big ideal rings in different contexts of commutative rings such us integrally closed domains, pullbacks and trivial ring extensions etc. We show that the notions of big and upper big ideals are completely different. The notion of big ideal is correlated to the notion of basic ideal and the notion of upper big ideal is correlated to the notion of $\mathcal{C}$-ideals. We give a new characterization of Pr\"ufer domains via big ideal domains and we characterize some particular cases of pullback rings that are big ideal domains. Also we give some classes of big and upper big ideals in rings with zero-divisors via trivial ring extensions.