Let $R$ be a commutative ring with identity, and let $M$ be aunitary 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 nontrivialexamples of HS modules of arbitrarily large cardinality overNoetherian and non-Noetherian domains. We then prove the followingresult: suppose $M$ is faithful over $R$, $R$ is a domain (we willshow 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, $Rsubseteq Msubseteq K$, andthere exists a generating set $S$ for $M$ over $R$ with $|S|<|R|$.We use this result to generalize a problem posed by Kaplansky andconclude the paper by answering an open question on Jónssonmodules.
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机译：假设$ R $是具有身份的交换环，并且让$ M $是$ R $之上的单一模块。当且仅当$ M $是无限的并且每个$ M $的非零子模块$ N $有$ | M / N | <| M | $时，我们称$ M $ H小（简称HS）。简要介绍之后，我们表明存在在Noetherian和非Noetherian域上任意大基数的HS模块的非平凡示例。然后，我们证明以下结果：假设$ M $是$ R $的忠实用户，$ R $是一个域（我们将证明我们可以在不损失一般性的情况下限制这种情况），而$ K $是$ R的商域$。如果$ M $是HS高于$ R $，则$ R $是HS作为其自身的模块，即$ Rsubseteq Msubseteq K $，并且存在$ R $且$ | S |的发电机组$ S $。 <| R | $。我们使用此结果来概括由Kaplansky提出的问题，并通过回答有关Jónssonmodules的公开问题来结束本文。
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