Abstract view
On Modules Whose Proper Homomorphic Images Are of Smaller Cardinality


Published:20110615
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
Abstract
Let $R$ be a commutative ring with identity, and let $M$ be a
unitary module over $R$. We call $M$ Hsmaller (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 nonNoetherian 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
