129 research outputs found
GCH implies the existence of many rigid almost free abelian groups
We begin with the existence of groups with trivial duals for cardinals
aleph_n (n in omega). Then we derive results about strongly aleph_n-free
abelian groups of cardinality aleph_n (n in omega) with prescribed free,
countable endomorphism ring. Finally we use combinatorial results of [Sh:108],
[Sh:141] to give similar answers for cardinals >aleph_omega. As in Magidor and
Shelah [MgSh:204], a paper concerned with the existence of kappa-free, non-free
abelian groups of cardinality kappa, the induction argument breaks down at
aleph_omega. Recall that aleph_omega is the first singular cardinal and such
groups of cardinality aleph_omega do not exist by the well-known Singular
Compactness Theorem (see [Sh:52])
Generalized E-Algebras via lambda-Calculus I
An R-algebra A is called E(R)-algebra if the canonical homomorphism from A to
the endomorphism algebra End_RA of the R-module {}_R A, taking any a in A to
the right multiplication a_r in End_R A by a is an isomorphism of algebras. In
this case {}_R A is called an E(R)-module. E(R)-algebras come up naturally in
various topics of algebra, so it's not surprising that they were investigated
thoroughly in the last decade. Despite some efforts it remained an open
question whether proper generalized E(R)-algebras exist. These are R-algebras A
isomorphic to End_R A but not under the above canonical isomorphism, so not
E(R)-algebras. This question was raised about 30 years ago (for R=Z) by Phil
Schultz and we will answer it. For PIDs R of characteristic 0 that are neither
quotient fields nor complete discrete valuation rings - we will establish the
existence of generalized E(R)-algebras. It can be shown that E(R)-algebras over
rings R that are complete discrete valuation rings or fields must trivial
(copies of R). The main tool is an interesting connection between
lambda-calculus (used in theoretical computer sciences) and algebra. It seems
reasonable to divide the work into two parts, in this paper we will work in V=L
(Godel universe) hence stronger combinatorial methods make the final arguments
more transparent. The proof based entirely on ordinary set theory (the axioms
of ZFC) will appear in a subsequent paper
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=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 rigid
tree-elements coming from a module generated over a forest of trees
Radicals and Plotkin's problem concerning geometrically equivalent groups
If G and X are groups and N is a normal subgroup of X, then the G-closure of
N in X is the normal subgroup X^G= bigcap{kerphi|phi:X-> G, with N subseteq
kerphi} of X . In particular, 1^G = R_G X is the G-radical of X. Plotkin calls
two groups G and H geometrically equivalent, written G H, if for any free group
F of finite rank and any normal subgroup N of F the G-closure and the H-closure
of N in F are the same. Quasiidentities are formulas of the form
(bigwedge_{i w =1) for any words w, w_i (i<=n) in a free group.
Generally geometrically equivalent groups satisfy the same quasiidentiies.
Plotkin showed that nilpotent groups G and H satisfy the same quasiidenties if
and only if G and H are geometrically equivalent. Hence he conjectured that
this might hold for any pair of groups. We provide a counterexample
Localizations of Groups
A group homomorphism eta:A-> H is called a localization of A if every
homomorphism phi:A-> H can be `extended uniquely' to a homomorphism Phi:H-> H
in the sense that Phi eta = phi. This categorical concepts, obviously not
depending on the notion of groups, extends classical localizations as known for
rings and modules. Moreover this setting has interesting applications in
homotopy theory. For localizations eta:A-> H of (almost) commutative structures
A often H resembles properties of A, e.g. size or satisfying certain systems of
equalities and non-equalities. Perhaps the best known example is that
localizations of finite abelian groups are finite abelian groups. This is no
longer the case if A is a finite (non-abelian) group. Libman showed that A_n->
SO_{n-1}(R) for a natural embedding of the alternating group A_n is a
localization if n even and n >= 10 . Answering an immediate question by Dror
Farjoun and assuming the generalized continuum hypothesis GCH we recently
showed in math.LO/9912191 that any non-abelian finite simple has arbitrarily
large localizations. In this paper we want to remove GCH so that the result
becomes valid in ordinary set theory. At the same time we want to generalize
the statement for a larger class of A 's
On Crawley Modules
This continues recent work in a paper by Corner, Gobel and Goldsmith. A
particular question was left open: Is it possible to carry over the results
concerning the undecidability of torsion--free Crawley groups to modules over
the ring of p-adic integers? We will confirm this and also strengthen one of
the older results in by replacing the hypothesis of diamondsuit by CH
Endomorphism rings of modules whose cardinality is cofinal to omega
The main result is
Theorem: Let A be an R-algebra, mu, lambda be cardinals such that
|A|<=mu=mu^{aleph_0}<lambda<=2^mu. If A is aleph_0-cotorsion-free or A is
countably free, respectively, then there exists an aleph_0-cotorsion-free or a
separable (reduced, torsion-free) R-module G respectively of cardinality
|G|=lambda with End_RG=A oplus Fin G
Characterizing automorphism groups of ordered abelian groups
We characterize the groups isomorphic to full automorphism groups of ordered
abelian groups. The result will follow from classical theorems on ordered
groups adding an argument from proofs used to realize rings as endomorphism
rings of abelian groups
Absolutely Indecomposable Modules
A module is called absolutely indecomposable if it is directly indecomposable
in every generic extension of the universe. We want to show the existence of
large abelian groups that are absolutely indecomposable. This will follow from
a more general result about R-modules over a large class of commutative rings R
with endomorphism ring R which remains the same when passing to a generic
extension of the universe. It turns out that `large' in this context has the
precise meaning, namely being smaller then the first omega-Erdos cardinal
defined below. We will first apply result on large rigid trees with a similar
property established by Shelah in 1982, and will prove the existence of related
` R_omega-modules' (R-modules with countably many distinguished submodules) and
finally pass to R-modules. The passage through R_omega-modules has the great
advantage that the proofs become very transparent essentially using a few
`linear algebra' arguments accessible also for graduate students. The result
gives a new construction of indecomposable modules in general using a counting
argument
Reflexive subgroups of the Baer-Specker group and Martin's axiom
In two recent papers (math.LO/0003164 and math.LO/0003165) we answered a
question raised in the book by Eklof and Mekler (p. 455, Problem 12) under the
set theoretical hypothesis of diamondsuit_{aleph_1} which holds in many models
of set theory, respectively of the special continuum hypothesis (CH). The
objects are reflexive modules over countable principal ideal domains R, which
are not fields. Following H.Bass, an R-module G is reflexive if the evaluation
map s:G ---> G^{**} is an isomorphism. Here G^*=Hom(G,R) denotes the dual
module of G. We proved the existence of reflexive R-modules G of infinite rank
with G not cong G+R, which provide (even essentially indecomposable) counter
examples to the question mentioned above. Is CH a necessary condition to find
`nasty' reflexive modules? In the last part of this paper we will show
(assuming the existence of supercompact cardinals) that large reflexive modules
always have large summands. So at least being essentially indecomposable needs
an additional set theoretic assumption. However the assumption need not be CH
as shown in the first part of this paper. We will use Martin's axiom to find
reflexive modules with the above decomposition which are submodules of the
Baer-Specker module R^omega
- …