On Modules Whose Proper Homomorphic Images Are of Smaller Cardinality

Canadian mathematical bulletin

Adam Salminen¹  Greg Oman² 

[1] Department of Mathematics, University of Evansville, Evansville, IN 47722, USA;Department of Mathematics, The University of Colorado at Colorado Springs, Colorado Springs, CO 80918, USA

关键词: Noetherian ring;    residually finite ring;    cardinal number;    continuum hypothesis;    valuation ring;    Jónsson module;   

DOI  :  10.4153/CMB-2011-120-0

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

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.

【 授权许可】

Unknown