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1. CMB 2011 (vol 55 pp. 378)

Oman, Greg; Salminen, Adam
 On Modules Whose Proper Homomorphic Images Are of Smaller Cardinality 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 JÃ³nsson modules. Keywords:Noetherian ring, residually finite ring, cardinal number, continuum hypothesis, valuation ring, JÃ³nsson moduleCategories:13A99, 13C05, 13E05, 03E50

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