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1. CJM 2003 (vol 55 pp. 750)
Almost-Free $E$-Rings of Cardinality $\aleph_1$ An $E$-ring is a unital ring $R$ such that every endomorphism of
the underlying abelian group $R^+$ is multiplication by some
ring element. The existence of almost-free $E$-rings of
cardinality greater than $2^{\aleph_0}$ is undecidable in $\ZFC$.
While they exist in G\"odel's universe, they do not exist in other
models of set theory. For a regular cardinal $\aleph_1 \leq
\lambda \leq 2^{\aleph_0}$ we construct $E$-rings of cardinality
$\lambda$ in $\ZFC$ which have $\aleph_1$-free additive structure.
For $\lambda=\aleph_1$ we therefore obtain the existence of
almost-free $E$-rings of cardinality $\aleph_1$ in $\ZFC$.
Keywords:$E$-rings, almost-free modules Categories:20K20, 20K30, 13B10, 13B25 |
2. CJM 1998 (vol 50 pp. 719)
Indecomposable almost free modules---the local case Let $R$ be a countable, principal ideal domain which is not a field and
$A$ be a countable $R$-algebra which is free as an $R$-module. Then we
will construct an $\aleph_1$-free $R$-module $G$ of rank $\aleph_1$
with endomorphism algebra End$_RG = A$. Clearly the result does not
hold for fields. Recall that an $R$-module is $\aleph_1$-free if all
its countable submodules are free, a condition closely related to
Pontryagin's theorem. This result has many consequences, depending on
the algebra $A$ in use. For instance, if we choose $A = R$, then
clearly $G$ is an indecomposable `almost free' module. The existence of
such modules was unknown for rings with only finitely many primes like
$R = \hbox{\Bbbvii Z}_{(p)}$, the integers localized at some prime $p$. The result
complements a classical realization theorem of Corner's showing that
any such algebra is an endomorphism algebra of some torsion-free,
reduced $R$-module $G$ of countable rank. Its proof is based on new
combinatorial-algebraic techniques related with what we call {\it rigid
tree-elements\/} coming from a module generated over a forest of trees.
Keywords:indecomposable modules of local rings, $\aleph_1$-free modules of rank $\aleph_1$, realizing rings as endomorphism rings Categories:20K20, 20K26, 20K30, 13C10 |