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1. CMB 2011 (vol 55 pp. 378)
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 module Categories:13A99, 13C05, 13E05, 03E50 |
2. CMB 2009 (vol 52 pp. 303)
A Comment on ``$\mathfrak{p} < \mathfrak{t}$'' Dealing with the cardinal invariants ${\mathfrak p}$ and
${\mathfrak t}$ of the continuum, we prove that
${\mathfrak m}={\mathfrak p} = \aleph_2\ \Rightarrow\ {\mathfrak t} =\aleph_2$.
In other words, if ${\bf MA}_{\aleph_1}$ (or a weak version of
this) holds, then (of course $\aleph_2\le {\mathfrak p}\le
{\mathfrak t}$ and) ${\mathfrak p}=\aleph_2\ \Rightarrow\
{\mathfrak p}={\mathfrak t}$. The proof is based on a criterion
for ${\mathfrak p}<{\mathfrak t}$.
Categories:03E17, 03E05, 03E50 |