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2018 Residually small commutative rings
Greg Oman, Adam Salminen
J. Commut. Algebra 10(2): 187-211 (2018). DOI: 10.1216/JCA-2018-10-2-187

Abstract

Let $R$ be a ring. Following the literature, $R$ is called \textit {residually finite} if, for every $r\in R\backslash \{0\}$, there exists an ideal $I_r$ of $R$ such that $r\notin I_r$ and $R/I_r$ is finite. In this note, we define a \textit {strictly} infinite commutative ring $R$ with identity to be \textit {residually small} if, for every $r\in R\backslash \{0\}$, there exists an ideal $I_r$ of $R$ such that $r\notin I_r$ and $|R/I_r|\lt |R|$. The purpose of this article is to study such rings, extending results on (infinite) residually finite rings.

Citation

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Greg Oman. Adam Salminen. "Residually small commutative rings." J. Commut. Algebra 10 (2) 187 - 211, 2018. https://doi.org/10.1216/JCA-2018-10-2-187

Information

Published: 2018
First available in Project Euclid: 13 August 2018

zbMATH: 06917493
MathSciNet: MR3842334
Digital Object Identifier: 10.1216/JCA-2018-10-2-187

Subjects:
Primary: 13A15
Secondary: 03E10

Keywords: Artinian ring , cofinality , homomorphically smaller ring , Noetherian ring , quotient ring , regular cardinal , residually finite ring

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.10 • No. 2 • 2018
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