Residually small commutative rings
type: Item
instance of: scholarly article
title: Residually small commutative rings
zbMATH Open document ID: 1400.13010
DOI: 10.1216/JCA-2018-10-2-187
author: Adam Salminen
author: Greg Oman
published in: Journal of Commutative Algebra
publication date: 2018-08-14T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jca/1534125825
review text: Well established in the literature is a property called residually finite which originated with groups, and since has been studied in modules and rings. A commutative ring \(R\) is called residually finite if for every \(r \in R\setminus \{0\}=R^*\), there exists an ideal \(I_r\) such that \(r\notin I_r\) and \(R/I_r\) is finite. This paper studies a related property of commutative rings called residually small. An infinite commutative ring \(R\) is said to be residually small if for every non-zero \(r \in R\), there exists and ideal \(I_r\) such that \(r\notin I_r\) and \(|R/I_r|< |R|\). It is worth noting that all infinite residually finite rings are therefore also residually small with this definition. This property is related to a property for modules which is called homomorphically smaller (HS) which means for an infinite module \(M\), \(|M/N|<|M|\) for every non-zero submodule \(N\) of \(M\). Then a ring \(R\) is HS if it is an HS module over itself. The paper begins with an extensive preliminary section which explores many of the properties of residually small rings and especially compares these properties with other known properties in the literature. Then significant attention is turned towards studying how the residually small property behaves with respect to polynomial ring extensions, direct products, quotient rings, factor rings and integral extensions. Finally, the authors conclude by posing several interesting open questions of which we include a few. (1) If \(R\) is a Noetherian local ring and \(I\) is an ideal of \(R\) with \(|R/I|=|R|\), then \(R\) is residually small if and only if \(R/I\) is residually small. Does this still hold if \(R\) is not Noetherian? (2) Suppose that \(D_1 \subseteq D_2\) is an integral extension of domains and that \(D_1\) is residually small and uncountable. Must \(D_2\) be residually small? Is this question even decidable in ZFC? What if the extension were finite?
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 03E10
zbMATH DE Number: 6917493
zbMATH Keywords: Artinian ring
zbMATH Keywords: cofinality
zbMATH Keywords: homomorphically smaller ring
zbMATH Keywords: Noetherian ring
zbMATH Keywords: quotient ring
zbMATH Keywords: regular cardinal
zbMATH Keywords: residually finite ring
reviewed by: Christopher P. Mooney
MaRDI profile type: Publication
cites work: The Set of Torsion Elements of a Module
cites work: Zero-Divisors, Torsion Elements, and Unions of Annihilators
cites work: Automorphism Groups of Residually Finite Groups
cites work: Residually Finite Rings
cites work: Note on residually finite rings
cites work: Q3137901
cites work: Profinite and residually finite groups
cites work: Q4873059
cites work: Q3530874
cites work: Cardinalities of Residue Fields of Noetherian Integral Domains
cites work: Rings with Finite Norm Property
cites work: Subrings of finite index in finitely generated rings
cites work: Residually finite groups
cites work: Jónsson Modules over Noetherian Rings
cites work: SMALL AND LARGE IDEALS OF AN ASSOCIATIVE RING
cites work: On modules whose proper homomorphic images are of smaller cardinality
cites work: Rings Which Admit Faithful Torsion Modules
cites work: Rings which admit faithful torsion modules. II.
cites work: Residual finiteness and the Hopf property in rings
cites work: Q5749462
cites work: Q3413553
cites work: Q4277307
cites work: Rings with all modules residually finite
OpenAlex ID: W2885455152
description: scientific article; zbMATH DE number 6917493