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On Modules Whose Proper Homomorphic Images Are of Smaller Cardinality

  Published:2011-06-15
 Printed: Jun 2012
  • Greg Oman,
    Department of Mathematics, The University of Colorado at Colorado Springs, Colorado Springs, CO 80918, USA
  • Adam Salminen,
    Department of Mathematics, University of Evansville, Evansville, IN 47722, USA
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Abstract

Let $R$ be a commutative ring with identity, and let $M$ be a unitary module over $R$. We call $M$ H-smaller (HS for short) if and only if $M$ is infinite and $|M/N|<|M|$ 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 HS over $R$, then $R$ is HS as a module over itself, $R\subseteq M\subseteq K$, and there exists a generating set $S$ for $M$ over $R$ with $|S|<|R|$. We use this result to generalize a problem posed by Kaplansky and conclude the paper by answering an open question on Jónsson modules.
Keywords: Noetherian ring, residually finite ring, cardinal number, continuum hypothesis, valuation ring, Jónsson module Noetherian ring, residually finite ring, cardinal number, continuum hypothesis, valuation ring, Jónsson module
MSC Classifications: 13A99, 13C05, 13E05, 03E50 show english descriptions None of the above, but in this section
Structure, classification theorems
Noetherian rings and modules
Continuum hypothesis and Martin's axiom [See also 03E57]
13A99 - None of the above, but in this section
13C05 - Structure, classification theorems
13E05 - Noetherian rings and modules
03E50 - Continuum hypothesis and Martin's axiom [See also 03E57]
 

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