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On Modules Whose Proper Homomorphic Images Are of Smaller Cardinality

Published online by Cambridge University Press:  20 November 2018

Greg Oman  and
Adam Salminen
Greg Oman
Affiliation:
Department of Mathematics, The University of Colorado at Colorado Springs, Colorado Springs, CO 80918, USAe-mail: goman@uccs.edu
Adam Salminen
Affiliation:
Department of Mathematics, University of Evansville, Evansville, IN 47722, USAe-mail: as341@evansville.edu
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Abstract

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Let $R$ be a commutative ring with identity, and let $M$ be a unitary module over $R$. We call $M$$\text{H}$-smaller ($\text{HS}$ for short) if and only if $M$ is infinite and $\left| M/N \right|\,<\,\,\left| M \right|$ for every nonzero submodule $N$ of $M$. After a brief introduction, we show that there exist nontrivial examples of HS modules of arbitrarily large cardinality over Noetherian and non-Noetherian domains. We then prove the following result: suppose $M$ is faithful over $R$, $R$ is a domain (we will show that we can restrict to this case without loss of generality), and $K$ is the quotient field of $R$. If $M$ is $\text{HS}$ over $R$, then $R$ is $\text{HS}$ as a module over itself, $R\,\subseteq \,M\,\subseteq \,K$, and there exists a generating set $S$ for $M$ over $R$ with $\left| S \right|\,<\,\left| R \right|$. We use this result to generalize a problem posed by Kaplansky and conclude the paper by answering an open question on Jónsson modules.

Keywords

13A9913C0513E0503E50Noetherian ringresidually finite ringcardinal numbercontinuum hypothesisvaluation ringJónsson module
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 2 , 01 June 2012 , pp. 378 - 389
Copyright
Copyright © Canadian Mathematical Society 2012

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