On Modules Whose Proper Homomorphic Images Are of Smaller Cardinality

Oman, Greg ; Salminen, Adam

Canadian mathematical bulletin, Tome 55 (2012) no. 2, pp. 378-389

Résumé

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.

Détail

DOI : 10.4153/CMB-2011-120-0

Mots-clés : 13A99, 13C05, 13E05, 03E50, Noetherian ring, residually finite ring, cardinal number, continuum hypothesis, valuation ring, Jónsson module