Abstract view
Indecomposable almost free modulesthe local case


Published:19980801
Printed: Aug 1998
Rüdiger Göbel
Saharon Shelah
Abstract
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 torsionfree,
reduced $R$module $G$ of countable rank. Its proof is based on new
combinatorialalgebraic techniques related with what we call {\it rigid
treeelements\/} coming from a module generated over a forest of trees.
MSC Classifications: 
20K20, 20K26, 20K30, 13C10 show english descriptions
Torsionfree groups, infinite rank unknown classification 20K26 Automorphisms, homomorphisms, endomorphisms, etc. Projective and free modules and ideals [See also 19A13]
20K20  Torsionfree groups, infinite rank 20K26  unknown classification 20K26 20K30  Automorphisms, homomorphisms, endomorphisms, etc. 13C10  Projective and free modules and ideals [See also 19A13]
