Rudiger Gobel, Daniel Herden and Saharon Shelah- Absolute E-rings

8/3/2019 Rudiger Gobel, Daniel Herden and Saharon Shelah- Absolute E-rings

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Absolute E-rings

Rudiger Gobel, Daniel Herden and Saharon Shelah

Abstract

A ring R with 1 is called an E-ring if EndZ R is ring-isomorphic to R under

the canonical homomorphism taking the value 1 for any EndZ R. MoreoverR is an absolute E-ring if it remains an E-ring in every generic extension ofthe universe. E-rings are an important tool for algebraic topology as explainedin the introduction. The existence of an E-ring R of each cardinality of theform 0 was shown by Dugas, Mader and Vinsonhaler [9]. We want to showthe existence of absolute E-rings. It turns out that there is a precise cardinal-barrier () for this problem: (The cardinal () is the first -Erdos cardinaldefined in the introduction. It is a relative of measurable cardinals.) We willconstruct absolute E-rings of any size < (). But there are no absolute E-rings of cardinality (). The non-existence of huge, absolute E-rings ()follows from a recent paper by Herden and Shelah [25] and the construction

of absolute E-rings R is based on an old result by Shelah [33] where familiesof absolute, rigid colored trees (with no automorphism between any distinctmembers) are constructed. We plant these trees into our potential E-rings withthe aim to prevent unwanted endomorphisms of their additive group to survive.Endomorphisms will recognize the trees which will have branches infinitely oftendivisible by primes. Our main result provides the existence of absolute E-ringsfor all infinite cardinals < (), i.e. these E-rings remain E-rings in all genericextensions of the universe (e.g. using forcing arguments). Indeed all previouslyknown E-rings ([9, 24]) of cardinality 20 have a free additive group R+ in someextended universe, thus are no longer E-rings, as explained in the introduction.

0

This is GbHSh 948 in the third authors list of publications.The collaboration was supported by the project No. I-963-98.6/2007 of the German-Israeli Foundationfor Scientific Research & Development and the Minerva Foundation.AMS subject classification: primary: 13C05, 13C10, 13C13, 20K20, 20K25, 20K30; secondary: 03E05,03E35. Key words and phrases: E-rings, tree constructions, absolutely rigid trees, indecomposableabelian groups

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Our construction also fills all cardinal-gaps of the earlier constructions (which

have only sizes 0). These E-rings are domains and as a by-product we obtainthe existence of absolutely indecomposable abelian groups, compare [23].

1 Introduction

We want to investigate E-rings and their absolute behavior. E-rings appeared whilestudying rings R with the property that the endomorphism ring EndZ R of the un-derlying additive structure is ring-isomorphic to R. (These rings are now called gen-eralized E-rings.) However, Schultz [32] was able to isolate in 1973 an importantclass of rings which since then are called E-rings: R is an E-ring if the evalua-tion map EndZ R R ( 1) is an isomorphism. (The name E-ring refers tothis particular mapping.) E-rings can also be defined dually: The homomorphismR EndZ R (r r) (with r scalar multiplication by r R on the right) isan isomorphism. Moreover, it is not hard to see that R is an E-ring if and only ifEndZR = R and R is commutative; see [24, pp. 468, 469 Proposition 13.1.9]. ThusR is an E-ring if and only if it is a commutative generalized E-ring. (This, of course,suggests the question about the existence of proper generalized E-rings, first noticed50 years ago by Fuchs [15] and answered recently by providing (in ordinary set theory,ZFC) the existence of a proper class of such non-commutative rings in [22].) The firstexamples of E-rings are the 20 subrings ofQ.

The class of E-rings was in the focus of many papers since then. The algebraicproperties were considered in fundamental papers by Mader, Pierce and Vinsonhaler[28, 30, 31] and the existence of arbitrarily large E-rings was first shown by examplesof rank 0 in Faticoni [12] (extended to ranks 2

0 in [24, p. 471, Corollary 13.2.3])and above 20 in Dugas, Mader, Vinsonhaler [9] using Shelahs Black Box as outlinedin Corner, Gobel [4]. The existence of related E-modules as a natural by-productappeared soon after in [7]. From [32] also follows that the torsion-part of an E-ringcan be classified; the same holds for the cotorsion-part as shown in [18]. In contrastthe quotients of the ring modulo the ideal of torsion-elements and the ideal generatedby the cotorsion submodules can be arbitrarily large as shown in [1, 18], respectively.

The existence ofE-rings contributes to algebraic topology: We rephrase the defini-tion by the diagram

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Z //

R

R,

where is the inclusion 1 R and for any there is a unique such that the diagramholds. However, this is the definition of a localization R of Z, see [3]. This notionmakes sense in many categories, and in particular can be studied in homotopy theory,as discussed in Dror Farjoun [5]. He raised the question if for a fixed compact space X,the distinct homotopy types of the form LfX form a set, where f : Y Z is runningthrough all possible maps between topological spaces and Lf denotes homotopical

localization with respect to these maps f. The following result is not hard to see, butis an important observation in the context of localizations of abelian Eilenberg-MacLane spaces. It will appear in Casacuberta, Rodrguez, Tai [3]: If a space X is ahomotopical localization of the circle S1 (i.e. X = LfS1), then X is the Eilenberg-Mac Lane space K(R, 1) with R an E-ring and any E-ring appears this way (takef : S1 K(R, 1) induced by the inclusion of 1 into R). (The Eilenberg-Mac Lanespace K(R, 1) is the connected space which has (abelian) fundamental group R andtrivial higher homotopy groups. It is unique up to homotopy and it is well-knownhow to construct such cellular models.) Thus the existence of a proper class of E-rings provides a negative answer to Dror Farjouns question. Below we will discuss an

absolute version of this result.Note that E-rings constructed earlier and here have also impact to other areas ofalgebra. They are useful for constructing nilpotent groups of class 2 (see Dugas, Gobel[8]) and build the core for investigating abelian groups with automorphism groupsacting uniquely transitive, see [19, 20, 21]. Surveys and classical results on E-rings canbe found in [13, 14, 24, 34].

The second ingredient of this paper is the notion of absolute structures. The recentactivity on this topic was initiated by Eklof and Shelah [11], who studied the existenceof absolutely indecomposable abelian groups. Here a property of a structure is calledabsolute if it is preserved under generic extensions of the given universe (of set theory),in particular it is preserved under forcing. Absolute formulas are discussed in detail ina classical monograph by Levy [27], examples are the subset relation, or the propertyto be an ordinal. A quick survey on absolute formulas is given in [2, pp. 408 412]. However, the powerset relation is not absolute. Here is a more striking algebraiccounterexample. The following statement (i) is not absolute.

(i) A = Z is an indecomposable abelian group and its subgroups of finite rank are

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free.

First we note that the freeness condition by Pontryagins theorem (Fuchs [16, Vol. 1,p. 93]) is equivalent to say that all countable subgroups of A are free, i.e. A is 1-free. We can find a generic extension of the underlying model of set theory (the Levycollapse) such that | A | becomes countable, hence A = Z is free and definitely notindecomposable. We immediately note, that all E-rings constructed in the past (andof size 20) are 1-free and thus can be treated the same way. They become free inan extended model and thus are no longer E-rings. The problem settled in this paperbecomes obvious.

Can we find absolute E-rings?

As a by-product of these considerations we obtain new, very useful methods forthe construction of complicated structures. The crucial point is, that often the oldconstructions use stationary sets or tools which are not that friendly from a constructivepoint of view: the new methods are based on inductive arguments and thus provide amore elementary approach to the desired complicated structures.

Surprisingly, there is a precise cardinal bound () for the construction of absoluteE-rings. Here () denotes the first -Erdos cardinal defined in Section 2. We noteimmediately that () (like the first measurable cardinal) is a large inaccessible cardinalwhich may not exist in any universe; see [26]. Any model of set theory contains asubmodel of ZFC which has no first -Erdos cardinal and it is also well-known thatGodels universe has no first -Erdos cardinal. In a recent paper Herden, Shelah [25]

have shown that there are no absolute E-rings of size (). We want to prove theconverse.

Main Theorem 1.1 If is any infinite cardinal< (), (the first-Erdos cardinal),then there is an absolute E-ring R of cardinality . MoreoverZ[X] R Q[X] withX a family of commuting free variables.

The new method of constructing E-rings differs from those described in the refer-ences and above. For example, the construction in [9] (which does not provide anyabsolute E-rings) - due to the Black Box - also does not allow to show the existenceof E-rings of cardinality cofinal with . However, Theorem 1.1 gives an answer for

all infinite cardinals < (). In Corollary 5.2 we explain how to extend this result toobtain rigid families of (absolute) E-rings.

The following application to algebraic topology is immediate by the above remarks.

Corollary 1.2 The familyLfS1 ( for any map f) of absolute localizations of the circle

S1 (based on Theorem 1.1) is a proper class, if and only if there is no -Erdos cardinal.

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Thus, in models of ZFC without -Erdos cardinals the negative answer to Dror

Farjouns problem is absolute.Some absolute constructions for other categories of modules, trees and graphs can be

seen in [23, 17, 33, 6]. In these cases it also follows that the upper bound () is sharp.However, it is still an open problem, if for the family of absolutely indecomposableabelian groups the upper bound can be larger than (), see also [11]. The strategyfor the construction of absolute E-rings utilizes the existence of absolutely rigid, coloredtrees from Shelah [33], which we will describe in Section 2. In fact, in order to applythis to E-rings, we first must strengthen [33] in Theorem 2.8.

Finally we explain the strategy of this paper in the simpler case of Theorem 1.1when X is (non-empty and) countable. In this case we can replace the existence of

absolutely rigid trees by a countable family of primes automatically resulting in anabsolute construction. Consider the family F = {x z, xn | x X, z Z, 0 < n .

Theorem 2.1 If < () is infinite and T = >, then there is a family (T, c)( < 2) of -colored subtrees of T (of size ) such that for , < 2 and in anygeneric extension of the universe the following holds.

Hom((T, c), (T, c)) = = = .

Remark 2.2 Such a family of colored trees (T, c) ( < 2) is called an absolutely

rigid family of trees of size . In the following we will show how to implement such afamily to construct absolute E-rings of any infinite cardinality < (). For > ()such an absolutely rigid family of trees does not exist.

We fix such a family and write

(T , ca) ( < 2

) for an absolutely rigid family of trees ( for a fixed < ()). (2.1)

2.1 A shift map for trees

In order to modify the family (2.1) we introduce two coding maps, which are bijections.

cd : >

andcd :

>( {}) ,

where denotes a new symbol (which does not appear in the set ).If < 2, then let := cd

1 () and define a subset T

> consisting of allelements > satisfying to the following two conditions. We let lg = n.

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(a) For < n let lg () = + 1.

(b) For any < n there is T such that

(+m)() =

(m) for m < lg ,

for lg m n 1.

This in particular implies that lg n 1 must hold. Given , then the choiceof elements is illustrated by the following diagram.

(0) (1) (2) (3) (n-1)

...

...

...

...

...

...

(0)

(1)

(2)

(3)

0

1

2

3

(n-1)

n-1

...

The triangular shape of the diagram is a direct consequence of the above conditions(a) and (b). The th line of the diagram is of the form

, , , . . . , where theelement T

is uniquely determined by and condition (b). Conversely, any choice

of elements T ( < n) with lg n 1 by (b) determines some T

of

length n with = ( < n), thus T = . We get an immediate

Proposition 2.3 Let 0 < (). The above set T is a colored subtree of>

with the coloring map

c : T , c

() = cd( lg | < lg

c() | < lg c() ).

Hence we have a family (T, c) | < 2

of colored trees.

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The relevant point here is that the color c() encodes the length and the color of

the branches for all < lg .

Proof. We have T = from above. Let T be an initial segment. We

must show that satisfies the two conditions (a),(b) above. Condition (a) is obvious.Condition (b) is satisfied with

=

(lg

1) for lg 1 < lg , otherwise.

Hence T. It is clear that c defines a coloring of T

.

In particular, holds for T and < lg

. Finally we illustrate the

above coloring c() for a tree.

(0) (1) (2) (3) (n-1)

...

...

...

...

...

...

(0)

(1)

(2)

(3)

0

1

2

3

(n-1)

n-1

...

0

1

2

3

n-1lg( )

lg(

lg(

lg(

lg( )

)

)

)

0

1

2

3

n-1c''( )

c''(

c''(

c''(

c''(

)

)

)

)

c''( )

c'( )

Next we will show that these trees are strongly rigid (in the sense of Theorem 2.5below). We will use the following natural definition.

Definition 2.4 If < 2 and T, then let (T) := { T

| } be the part

of the tree T

above .Using Theorem 2.1 we will establish the following

Theorem 2.5 If , < 2 are distinct, and T, then there is no color preservingpartial tree homomorphism : (T) T

in any generic extension of the universe.

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Proof. Suppose for contradiction that : (T) T is a color preserving

partial tree homomorphism. First we define an injective projection map

: T (T),

which we then want to compose with . If T and < lg , then we determine abranch

>( {}) of length lg = + lg + 1. Let

(m) :=

() for m = lg ,

otherwise.

Then put () = cd() | < lg which belongs to (T

) as required: Condition

(a) above is clear and (b) can be seen directly from the diagram below. Thus lg () =lg + lg , and (), lg = holds for = . Moreover, if, then also ()().

So : T (T) is an injective map that preserves initial segments.

Finally we want to define a color preserving tree homomorphism

: T T . (2.2)

by setting() = ()(), lg

for = and () = . Note that () is well-defined: For this we must show that

lg < lg(

)() for = . But note that by the above (using that

preserves length)

lg()() = lg () = lg + lg > lg

as desired.

() in (T' )

( )() in T''

'

'

lg lg

( )(),lg

()

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It is clear that () T . Finally we have to show that preserves the length and

color of branches as well as initial segments.If T then by the properties of mentioned above we have () (

) (T), and using that

is a tree homomorphism also ()() ()() and

() = ()(), lg ()(), lg = ()

holds. To show that preserves length and color we recall c(()) = c((

)()) for T as

preserves colors. However c codes the length and color of the elementsof the form ... lg (if = ) and it follows by definition of and , respectively, that

lg = lg (), lg = lg ()(), lg = lg ()

and similarly c() = c(()). As c

always codes the color of the root we also havec() = c

() = c

(()), while lg = lg () is obvious.

Hence is a color preserving tree homomorphism, which by Theorem 2.1 can notexist unless = . This case however was excluded.

2.2 Strongly rigid trees

In the final step of the tree construction we will modify the trees from Section 2.1 toprove the non-existence of color preserving partial tree homomorphisms on an even

smaller domain. It helps to consider for branches >

and >

with lg = lg the induced branch

:= () + () | < lg >( ) = >.

If >, then there is an obviously unique decomposition = with >, > and lg = lg . Furthermore, 1 1

2 2 holds iff

1

2 and 1 2.

Using the trees T ( < 2) from Theorem 2.5 we put

T := { > | = , T,

> and lg = lg }

and define a coloring

c() = cd(c(

) | < lg ) for = T.

Here cd is the coding map from the beginning of Section 2.1.Our final tree-results are the following two theorems.

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Theorem 2.6 Let(T, c) ( < 2) be as above. Then the following holds.

(i) T > is a subtree.

(ii) c : T is a coloring.

(iii) For T and T with c() = c() follows

(a) lg = lg .

(b) c( ) = c( ) for all < lg .

(c) If = , = then = .

Proof. It is clear that T = , and conditions (i) and (ii) are obvious. For (iii) weconsider c() = c(). Thus c

(

) | < lg ) = c( ) | < lg ) by

definition of the coloring. We get lg = lg = lg = lg , = and c( ) =

c( ) for all < lg . Now (a),(b) and (c) are obvious.

In preparation of the next theorem we define a special closure property.

Definition 2.7 We will also use the following closure condition for subsets T Tand = T :

(1) If =

T

then

(T

)

.(2) If = T and

T and lg

= lg + 1, then there is >with lg = lg + 1 and T .

Theorem 2.8 If (T, c) ( < 2) is as above and T T satisfies the closure

condition from Definition 2.7 for = T and = < 2, then there is no

color preserving partial tree homomorphism T T in any generic extension of theuniverse.

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Proof. Let = be as in the theorem and suppose for contradiction that

: T T is a color preserving partial tree homomorphism. We want to define acolor preserving partial tree homomorphism

: (T) T.

In the first step we define recursively a partial tree homomorphism

g : (T) >

such that g() T for all (T) . The (relative) bottom element is

(T) and we put g(

) = and note that = T by assumption of the

theorem. For the inductive step we consider

(T

)

,

g(

) T

, let

bewith lg = lg + 1 and define g() with the help of Definition 2.7(2). In particularg() g() and lg g() = lg g() + 1 = lg + 1. Hence g is well-defined on (T)and preserves lengths and initial segments.

Recall that for any (T) we have = g() T . In particular,

() = T is well-defined, and since preserves colors, we derive fromTheorem 2.6(iii)(c) that = g(); hence

( g()) = g() (2.3)

and we put

(

) =

T

. Thus the map

above is defined and we must checkthat it preserves initial segments, lengths and colors.

Let and recall that g preserves initial segments. Hence also g() g()and g() g(), and since is a partial tree homomorphism we conclude() g() () g() and () () from (2.3).

From ( g()) = () g() and the assumption that preserves colors[together with Theorem 2.6(iii)(b),(c)] we get c(

) = c(()) and see that also

preserves colors. Moreover lg = lg( g()) = lg(() g()) = lg () and also preserves the length. Such a map however is forbidden by Theorem 2.5 for = , so Theorem 2.8 holds.

3 The construction of E-rings

Let < () be a fixed infinite cardinal and enumerate by

= {pnki, qnki | n, k,i < }

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some of the primes ofZ without repetition. Let Q denote the field of rational numbers.

If p and a is an element of a torsion-free abelian group M, then we denote (asusual) by pa the family of unique elements {pna | n < } of the divisible hullQM = Q M using M QM. If pa M, we will also write p | a (in M).

First we decompose into =

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4 Invariant principal ideals of R

Theorem 4.1 If EndZ R and r R, then r (Rr).

Here (Rr) denotes the (unique) group purification of the principal ideal Rr of R,which is the smallest ideal I of R containing Rr with torsion-free abelian quotientR/I. Theorem 4.1 can be rephrased saying that all purified principal ideals of R arefully invariant under the action of EndZ R.

Recall that a submodule U of an R-module M is fully invariant ifU is an EndR M-submodule of M. We begin with a countable family of ideals which by arithmeticalreasons are obviously fully invariant ideals of the ring R:

If p = pnki , then Inki := pR =


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Proof. The inclusion R(x x) | Un, T Jnki holds because Jnki is

pure in R. Next we show that Jnki ker Fqnki:Note that (x x)F

qnki = 0 whenever Un, c() = i, lg = k + 1 and = k,

thusRFqnki p

Z[X] | qnki = p Q[X] .

If r Jnki, then r qnki R and rF

qnki q

nki (RF

qnki). But by the above inclusion

qnki (RFqnki) = 0 is obvious, hence r ker F

qnki.

For ker Fqnki R(x x) | Un, T we consider any 0 = r ker Fqnki.

As an element from R there is an integer a = 0 such that ar Z[X] can be expressedas a finite sum, where we isolate the variables x X that meet the restriction of thelemma. Thus we represent

ar =s

j=1

mj(x) fj + g,

where the monomials mj Z[X] contain only the x with c() = i, lg = k + 1, = k while the polynomials fj, g Z[X] do not have contributions from this set. Wecan express x = (x x) + x and rewrite the sum as

ar =

(x x) f + g

with suitable polynomials f Z[X] and g =s

j=1 mj(x) fj + g a polynomialwithout contributions from x. We now apply F

qnki and get

0 = (ar)Fqnki =

[(xx)f]F

qnki+g

Fqnki =

(xx)Fqnkif

F

qnki+g

Fqnki = g.

Thus g = 0 and ar =

(x x) f

R(x x) | Un, T. It follows

that r belongs to the corresponding purification as claimed. Thus the three displayedsets of the lemma coincide.

Similarly we can characterize the ideals Inki. The proof follows the arguments ofthe previous lemma.

Lemma 4.4 For p = pnki the following holds.

Inki = ker Fpnki = Rx | Un, T, c() = i, lg = k + 1 R.

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We now come to the proof of Theorem 4.1. Let EndZ R and 0 = r R be as

in the theorem. We fix some n and Un with r = rn = x and consider thefamily X = {x | T} of generators from R and let Y = {y := x | T}.From the definition of the variables x, the ideals Inki, Jnki and the observation thatthese ideals are fully invariant we get

x = r and y = r.

If lg > 0, then x X.

If c() = i, lg = k + 1, = k, then x Inki and x x Jnki.

If c() = i, lg = k + 1, = k, then y Inki and y y Jnki.The next definition helps to investigate Y.

Definition 4.5 If T, then let () be the set of all monomials from X whichappear in the canonical representation ofy inQ[X]. Ifm X, then letactive(m)be the list of all variables x from m with c() = c().

The definition of active(m) will mainly be used for m (). Note that a listis not a set: a variable x will appear with its multiplicity (for m) which in generalmay be > 1. We do not care about the ordering of this list.

Corollary 4.6 For T, c() = i, lg = k + 1 and m () the following holds.

(i) active(m) = .

(ii) If k > 0 and = k, then m := mFqnki X and for the lists we have(active(m))F

qnki active(m

).

From Corollary 4.6(ii) follows immediately | active(m) | | active(m) | for thesizes of the lists.

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Proof. (i) From x Inki follows y Inki and by Lemma 4.4 we have yFpnki = 0,

hence also mFpnki = 0. Note that the members of active(m) have color c() = i andthus active(m) = as the map F

pnki replaces x active(m) X with 0.

(ii) If x active(m), then lg = lg = k + 1 by Theorem 2.6(iii)(a) and the

map Fqnki replaces x by x X (as k > 0) with = k. Thus mFqnki X.

Furthermore, from x active(m) follows c() = c() and c(

) = c()by Theorem 2.6(iii)(b). Thus x = xF

qnki active(m

) and for the lists we have(active(m))F

qnki active(m

).

Recall the trees T, T from Section 2.

Corollary 4.7 Let = T with lg = k and T with lg

= k + 1.

Then there is some branch with the following properties.(i) > with lg = k + 1.

(ii) If x X appears in the canonical representation of y with = , then

= .

(iii) If := T for some T with lg

= k + 1, then c() = c() for allx X which appear in the canonical representation of y.

Proof. On the one hand there are only finitely many x which may appear in y,on the other hand there are infinitely many choices for

> with (i). So it is

easy to choose with (ii). Property (iii) is an immediate consequence of (i) and (ii):The branch T is well-defined by (i) and the definition of T, while fromc() = c() follows = by Theorem 2.6(iii)(c), contradicting (ii).

Definition 4.8 If = T with lg = k, then set

suc() = { = |

T, lg

= k + 1}

as a set of special successors of and

T = { T | ( + 1) suc( ) for all < lg }

as the subtree of T induced by these successors.

The next corollary shows that for any T

the set (T

) satisfies the closurecondition from Definition 2.7 and thus qualifies for Theorem 2.8.

Corollary 4.9 If = T , then (T ) satisfies the closure condition from

Definition 2.7 for . In particular T > is a subtree with the closure condition for

= T .

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Proof. It is clear that T and (T ) are closed under initial segments, thus subtrees

of >. The closure condition is also immediate from Corollary 4.7. Note that = T with lg

= k and T with lg = k + 1 implies that has

successors in T, hence some exists and also suc() = .

Corollary 4.10 If T , c() = i, lg = k + 1 and = k, then the followingholds.

(a) Ifm (), then mFqnki = m.

(b) yFqnki = y.

(c) Ifk > 0, m (), then there is m

() with m

F

q

nki = m.

Proof. (a) Suppose that some x X appears in m which is not a fix-point ofFqnki. Then necessarily c() = i = c() which contradicts Corollary 4.7(iii).

(b) By the choice of x x Jnki we also have y y Jnki and thus(y y)F

qnki = 0 by Lemma 4.3. It follows 0 = yF

qnki yF

qnki = yF

qnki y

which is (b).(c) We write y =

mi()

aimi and y =

mj()

ajmj; by (b) follows

mj(

)

ajmjF

qnki = yF

qnki = y = m

i(

)

aimi .

The summands on the left hand side are monomials in X by Corollary 4.6(ii) andk > 0. Comparing the two sides, for any mi () there must be an m

j ()

with mjFqnki = mi. So mi = m demonstrates (c).

Definition 4.11 If T , c() = i, lg = k + 1, = k and k > 0. Then let

g : () () with mgFqnki = m for all m ().

The map g is well-defined by Corollary 4.10(c) and the following holds by Corol-lary 4.6(ii).

Proposition 4.12 If g is defined as in Definition 4.11, then

Fqnki active(mg) : active(mg) active(m)

is an injective map of the lists.

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The following innocent looking lemma collects most of the earlier results and is the

platform for the final stage of the proof of Theorem 4.1.

Lemma 4.13 If T , lg = k + 1 and m (), then there is T such thatx active(m).

Proof. If is as in the lemma, then we want to define inductively a family {m | (T )} with

(i) m (),

(ii) m := m,

(iii) m := mg for T with c(

) = i, lg < lg = k+1 and = k.

If m is from this list, then mFqnki = m. First we consider the family

{| active(m) | | (T )}

If12, then | active2(m2) | | active1(m1) | by Corollary 4.6(ii). Choose (T ) with | active(m) | minimal. Then | active(m) | is constant for all (T ). If now

T with c() = i, lg < lg = k + 1, = k, then

Fqnki : active(m ) active(m) is a bijection of lists. (4.1)

By Corollary 4.6(i) we also have that active(m) = . So we can choose

x active(m) (4.2)

and we define inductively a color preserving partial tree homomorphism

: (T ) T such that x() active(m).

First we choose () = T. Since x active(m) we get c() = c() and

preserves the color at this stage. Moreover, since the colors code the branches from>

and the lengths of branches, also lg

= lg and preserves the length at thisstage. In the inductive step we consider T with c(

) = i, lg < lg =k + 1, = k and () = T such that x active(m). By (4.1) there isx active(m) with xF

qnki = x . We put (

) = . By definition of Fqnkifollows = , and T with

(lg 1) = . Thus () T preserves lengthsand initial segments; moreover x() active (m). Finally x() active (m )

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implies that c(()) = c(

), so also preserves the color and thus is as required

above. We are ready to apply Theorem 2.8 (together with Corollary 4.9) and derivethat = . By (4.2) there is T such that x active(m). ApplyingCorollary 4.6(ii) and we also find some x active(m) = active(m) andthe crucial lemma is shown.

The final stage of the proof of Theorem 4.1. We now chose any T withlg = 1, c() = i. By Lemma 4.13 we can write

y =

mi()

aimi =

mi()

aiximi ,

where mi = ximi with xi active(mi). It follows that c(i) = c() andthus lg i = lg = 1. By the earlier choice of x = r, the definition of y andCorollary 4.10(b) we get from the above that

r = x = y = yFqn0i =

mi()

ai(xiFqn0i)(m

iF

qn0i) = r

mi()

ai(miF

qn0i)

is an element from (Rr).

5 The Main Theorem and Consequences

5.1 Proof of Main Theorem 3.1

Lemma 5.1 Let EndZQ[X]+ with

f Q[X] f for all f Q[X],

then is multiplication by an element ofQ[X].

Proof. By hypothesis on we find for each f Q[X] an element gf Q[X]such that f = f gf. If m X is a monomial and x X, then m = m

gm = m(x) gm(x) and (xm) = xm gxm = x m(x) gxm(x) seen as functions g(x)depending on x. Now we fix r Q and use EndZQ[X]+ = EndQQ[X]+ to compute(rm xm) = r m (xm) = r m(x) gm(x) x m(x) gxm(x), while by hypothesisalso (rm xm) = (rm xm) grmxm(x) holds. Thus

(rm xm) grmxm(x) = r m(x) gm(x) x m(x) gxm(x).

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We now substitute x := r into this polynomial equation and get

0 = r m(r) gm(r) r m(r) gxm(r) (5.1)

which holds for all r Q. If r = 0 also rm(r) is a non-zero element of the integraldomain Q[X], so (5.1) gives

h(r) = 0 for h(x) = gm(x) gxm(x) and for all 0 = r Q.

Thus x r is a factor of h(x) for infinitely many r Q, which is only possible ifh is the zero-polynomial and gm = gxm. We apply this recursively for all monomialsm X to get gm = g1 for all m X, and it is now clear (by linearity) that also

gf = g1 for all 0 = f Q[X]. We conclude = g1 id, where id denotes the identitymap on Q[X].

Proof of Main Theorem 3.1. Let EndZ R for the ring R constructed in Section 3.Since the additive group ofQ[X] is divisible, can be lifted to a group endomorphismofQ[X]+ and satisfies by Theorem 4.1 the hypothesis of Lemma 5.1. Thus = g idfor some polynomial g Q[X]. However 1 = g R which completes the proof.

5.2 Large families of E-rings

The Main Theorem 3.1 can easily be extended to a family of rigid E-rings. For thisdecompose the family of trees given by Theorem 2.1 into 2 families of trees {(T, c) | i} of size 2

(i < 2) and apply the earlier arguments for the correspondingfamilies of trees. We get E-rings Ri (i < 2

) and the following holds.

Corollary 5.2 If is any infinite cardinal< () (the first-Erdos cardinal), there isa familyRi (i < 2

) of absolute E-rings of cardinality. IfHomZ(R+i , R

+j ) = 0 in some

generic extension of the universe for some i,j < 2, then i = j; thus {Ri | i < 2} is

absolutely rigid, and also Z[X] Ri Q[X] for alli < 2 for a setX of commuting

free variables.

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Address of the authors:Rudiger Gobel, Daniel Herden,Fakultat fur Mathematik,Universitat Duisburg-Essen,Campus Essen, 45117 Essen, Germanye-mail: [email protected], [email protected]

Saharon Shelah,Einstein Institute of Mathematics,Hebrew University of Jerusalem,Givat Ram, Jerusalem 91904, Israel

andRutgers University,Department of Mathematics,Hill Center, Busch Campus,Piscataway, NJ 08854 8019, USAe-mail: [email protected]

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Rudiger Gobel, Daniel Herden and Saharon Shelah- Absolute E-rings