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) iff $M$ is infinite and $|M/N| <|Mj|$ 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 ⊆ M ⊆ 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 J´onsson modules.