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1. 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 |