QUASIREGULAR TORSION RINGS HAVING ISOMORPHIC ADDITIVE AND CIRCLE COMPOSITION GROUPS

Abstract

We investigate the role played by torsion properties in determining whether or not a commutative quasiregular ring has its additive and circle composition (or adjoint) groups isomorphic. We clarify and extend some results for nil rings, showing, in particular, that an arbitrary torsion nil ring has the isomorphic groups property if and only if the components from its primary decomposition into p-rings do too.

We look at the more specific case of finite rings, extending the work of others to show that a non-trivial ring with the isomorphic groups property can be constructed if the additive group has one of the following groups in its decomposition into cyclic groups: Z₂ ⁿ (for n ≥ 3), Z₂ ⊕ Z₂ ⊕ Z₂, Z₂ ⊕ Z₄, Z₄ ⊕ Z₄, Z _p ⊕ Z _p (for odd primes, p), or Z _p ⁿ (for odd primes, p, and n ≥ 2).

We consider, also, an example of a ring constructed on an infinite torsion group and use a specific case of this to show that the isomorphic groups property is not hereditary.     

When ∏ is Isomorphic to ⊕

For a category , we investigate the problem of when the coproduct ⊕ and the product functor ∏ from  ^I to  are isomorphic for a fixed set I, or, equivalently, when the two functors are Frobenius functors. We show that for an Ab category  this happens if and only if the set I is finite (and even in a much general case, if there is a morphism in  that is invertible with respect to addition). However, we show that ⊕ and ∏ are always isomorphic on a suitable subcategory of  ^I which is isomorphic to  ^I but is not a full subcategory. If  is only a preadditive category, then we give an example that shows that the two functors can be isomorphic for infinite sets I. For the module category case, we provide a different proof to display an interesting connection to the notion of Frobenius corings.     

Transfer and Tate’s theorem

Let H be a subgroup of a finite group G, and assume that p is a prime that does not divide |G : H|. In favorable circumstances, one can use transfer theory to deduce that the largest abelian p-groups that occur as factor groups of G and of H are isomorphic. When this happens, Tate’s theorem guarantees that the largest not-necessarily-abelian p-groups that occur as factor groups of G and H are isomorphic. Known proofs of Tate’s theorem involve cohomology or character theory, but in this paper, a new elementary proof is given. It is also shown that the largest abelian p-factor group of G is always isomorphic to a direct factor of the largest abelian p-factor group of H. Received: 17 June 2008     

Solvable absolute Galois groups are metabelian

We show that solvable absolute Galois groups have an abelian normal subgroup such that the quotient is the direct product of two finite cyclic and a torsion-free procyclic group. In particular, solvable absolute Galois groups are metabelian. Moreover, any field with solvable absolute Galois group G admits a non-trivial henselian valuation, unless each Sylow-subgroup of G is either procyclic or isomorphic to Z ₂⋊Z/2Z. A complete classification of solvable absolute Galois groups (up to isomorphism) is given. Oblatum 22-IV-1998 & 1-IX-2000?Published online: 30 October 2000     

On the Almost Sure Central Limit Theorem for a Class of Z d -Actions

As an extension of earlier papers on stationary sequences, a concept of weak dependence for strictly stationary random fields is introduced in terms of so-called homoclinic transformations. Under assumptions made within the framework of this concept a form of the almost sure central limit theorem (ASCLT) is established for random fields arising from a class of algebraic Z ^d -actions on compact abelian groups. As an auxillary result, the central limit theorem is proved via Ch. Stein's method. The next stage of the proof includes some estimates which are specific for ASCLT. Both steps are based on making use of homoclinic transformations.     

Sumset phenomenon in countable amenable groups

Jin proved that whenever A and B are sets of positive upper density in Z, A+B is piecewise syndetic. Jin's theorem was subsequently generalized by Jin and Keisler to a certain family of abelian groups, which in particular contains Zd. Answering a question of Jin and Keisler, we show that this result can be extended to countable amenable groups. Moreover we establish that such sumsets (or — depending on the notation — “product sets”) are piecewise Bohr, a result which for G=Z was proved by Bergelson, Furstenberg and Weiss. In the case of an abelian group G, we show that a set is piecewise Bohr if and only if it contains a sumset of two sets of positive upper Banach density.     

On infinite tournaments with regular automorphism groups

Atournament regular representation (TRR) of an abstract groupG is a tournamentT whose automorphism group is isomorphic toG and is a regular permutation group on the vertices ofT. L. Babai and W. Imrich have shown that every finite group of odd order exceptZ ₃ ×Z ₃ admits a TRR. In the present paper we give several sufficient conditions for an infinite groupG with no element of order 2 to admit a TRR. Among these are the following: (1)G is a cyclic extension byZ of a finitely generated group; (2)G is a cyclic extension byZ _2n+1 of any group admitting a TRR; (3)G is a finitely generated abelian group; (4)G is a countably generated abelian group whose torsion subgroup is finite.     

Strong Morita equivalence of higher-dimensional noncommutative tori. II

We show that two C*-algebraic noncommutative tori are strongly Morita equivalent if and only if they have isomorphic ordered K ₀-groups and centers, extending N. C. Phillips’s result in the case that the algebras are simple. This is also generalized to the twisted group C*-algebras of arbitrary finitely generated abelian groups. This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada, held by George A. Elliott.     

Rings consisting entirely of certain elements

We completely determine when a ring consists entirely of weak idempotents, units and nilpotents. We prove that such ring is exactly isomorphic to one of the following: a Boolean ring; Z₃ ⊕ Z₃; Z₃ ⊕ B where B is a Boolean ring; local ring with nil Jacobson radical; M₂(Z₂) or M₂(Z₃); or the ring of a Morita context with zero pairings where the underlying rings are Z₂ or Z₃.     

The relative Picard functor on schemes over a symmetric monoidal category

We consider schemes (X,OX) over an abelian closed symmetric monoidal category (C,⊗,1). Our aim is to extend a theorem of Kleiman on the relative Picard functor to schemes over (C,⊗,1). For this purpose, we also develop some basic theory on quasi-coherent modules on schemes (X,OX) over C.     

On the homeomorphism of continuous mappings

Denote by C(X) the partially ordered (PO) set of all continuous epimorphisms of a space X under the natural identification of homeomorphic epimorphisms. The following homeomorphism theorem for bicompacta is implicitly contained in Magill’s 1968 paper: two bicompacta X and Y are homeomorphic if and only if the PO sets C(X) and C(Y) are isomorphic. In the present paper, Magill’s theorem is extended to the category of mappings in which the role of bicompacta is played by perfect mappings. The results are obtained in two versions, namely, in the category TOP _Z (of triangular commutative diagrams) and in the category MAP (of quadrangular commutative diagrams).     

Torsion-free constructive nilpotent <Emphasis Type="Italic">R</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-groups

We consider a torsion-free nilpotent R _p -group, the p-rank of whose quotient by the commutant is equal to 1 and either the rank of the center by the commutant is infinite or the rank of the group by the commutant is finite. We prove that the group is constructivizable if and only if it is isomorphic to the central extension of some divisible torsion-free constructive abelian group by some torsion-free constructive abelian R _p -group with a computably enumerable basis and a computable system of commutators. We obtain similar criteria for groups of that type as well as divisible groups to be positively defined. We also obtain sufficient conditions for the constructivizability of positively defined groups.     

Weakly Stable Modules

ABSTRACT

A new notion which is called weakly stable module is introduced in this article. It is a nontrivial generalization of the modules with endomorphism rings having stable range one. We deduce that weakly stable projective modules have the cancellation property, and so any commutative hereditary ring has the cancellation property, i.e., if R is a commutative hereditary ring, then for any R-modules B and C, R ⊕ B ? R ⊕ C implies B ? C.     

On pure subgroups of locally compact abelian groups

In this note, we construct an example of a locally compact abelian group G = C × D (where C is a compact group and D is a discrete group) and a closed pure subgroup of G having nonpure annihilator in the Pontrjagin dual $\hat{G}$, answering a question raised by Hartman and Hulanicki. A simple proof of the following result is given: Suppose ${\frak K}$ is a class of locally compact abelian groups such that $G \in {\frak K}$ implies that $\hat{G} \in {\frak K}$ and nG is closed in G for each positive integer n. If H is a closed subgroup of a group $G \in {\frak K}$, then H is topologically pure in G exactly if the annihilator of H is topologically pure in $\hat{G}$. This result extends a theorem of Hartman and Hulanicki.Received: 4 April 2002     

Walker groups

A reformulation of Walker’s theorem on the cancellation of \(\mathbf {Z}\) says that any two homomorphisms from an abelian group W onto \(\mathbf {Z}\) have isomorphic kernels. It does not have a constructive proof, even for W a subgroup of \(\mathbf {Z}^{3}.\) In this paper we give a constructive proof of Walker’s theorem for W a direct sum, over any discrete index set, of groups of the following two kinds: Butler groups with weakly computable heights, and finite-rank torsion-free groups B with computable relative heights (that is, all quotients of B by finite-rank pure subgroups have computable heights). Throughout, “group” means abelian group. The infinite cyclic group, and the ring of integers, is denoted by \(\mathbf {Z}.\) The nonnegative integers are denoted by \(\mathbf {N},\) the positive integers by \(\mathbf {Z}^{+},\) and the rational numbers by \(\mathbf {Q}.\) We say that a set is discrete if any two elements are either equal or different. A subset A of a set B is detachable (from B) if for each \(b\in B,\) either \(b\in A\) or \(b\notin A.\) A group is discrete if and only if its subset \(\{0\}\) is detachable. Walker, in his dissertation [7], and Cohn in [2], showed that \(\mathbf {Z}\) is cancellable in the sense that if \(\mathbf {Z}\oplus B\cong \mathbf {Z}\oplus B^{\prime },\) then \(B\cong B^{\prime }.\) It is somewhat of an oddity that \(\mathbf {Z}\) is cancellable. A rank-one torsion-free group A is cancellable if and only if \(A\cong \mathbf {Z}\) or the endomorphism ring of A has stable range one [3], [1, Theorem 8.12]. (A ring R has stable range one if whenever \(aR+bR=R,\) then \(a+bR\) contains a unit of R.) In fact, any object in an abelian category whose endomorphism ring has stable range one is cancellable. The endomorphism ring of \(\mathbf {Z}\) does not have stable range one, so \(\mathbf {Z}\) is the unique rank-one torsion-free group that is cancellable for some reason other than its endomorphism ring. Walker’s theorem can be reformulated to say that any two maps from an abelian group W onto \(\mathbf {Z}\) have isomorphic kernels. Accordingly, we define a Walker group to be such a group W. Of course, Walker’s theorem says that every abelian group is a Walker group. However, a counterexample in the (abelian) category of diagrams \(\cdot \rightarrow \cdot \rightarrow \cdot \) of abelian groups provides a Kripke model which shows that there is no constructive proof that even every subgroup of \(\mathbf {Z}^{3}\) is a Walker group [4]. Thus, from a constructive point of view, it is of interest to explore the class of Walker groups. We will say that a group is a cZ-group if every homomorphism from it into \(\mathbf {Z}\) has a cyclic image. An easy classical argument shows that every group is a cZ-group. It is an immediate (constructive) consequence of [4, Theorem 1] that every cZ-group is a Walker group. This is not a complete triviality because it provides a classical proof of Walker’s theorem! The question remains as to how extensive the class of cZ-groups is. That question motivated the current paper. Our main results along these lines are Theorem 1.3, which says that Butler groups with weakly computable heights are cZ-groups, and Theorem 1.6, which says that a finite-rank torsion-free group with computable relative heights is a cZ-group (Butler groups with computable heights have computable relative heights). The relevance of the ability to compute heights to the study of Walker groups was suggested by the fact that this was not possible for the group corresponding to the counterexample. The notions of weakly computable heights and computable heights already appeared in [5, 6], papers that are over 20 years old. The notion of computable relative heights is stronger than these and originates in the current paper, just after Theorem 1.3. Note that B is a cZ-group if and only if \(\mathbf {Z}\oplus B\) is a cZ-group. Finitely generated groups are clearly cZ-groups. Finite direct sums of cZ-groups, and quotients of cZ-groups, are cZ-groups. As any map into \(\mathbf {Z}\) kills all torsion elements, we will focus on torsion-free groups B. However, not even nonzero subgroups of \(\mathbf {Z}\) need be cZ-groups: for example, \(\{x\in \mathbf {Z}:\,x\,\mathrm{is\,even,\, or}\,P\}.\) In Sect. 2 we show that a direct sum of cZ-groups over a discrete index set is a Walker group (Corollary 2.3). This gives essentially the largest class of Walker groups that we currently know (Corollary 2.4), although in [4, Theorem 5] it was shown that if B is a torsion-free group such that every nonzero map from B into \(\mathbf {Z}\) is one-to-one, then \(\mathbf {Z}\oplus B\) is a Walker group. Rank-one torsion-free groups B have that property, as do subgroups of \(\mathbf {Z},\) and any group with no nontrivial maps into \(\mathbf {Z}.\) The question regarding a group B that is a finite direct sum of such groups was left open, and is still open as far as I know. Section 3 deals with the idea of the height of a subgroup. This idea arose in an effort to formulate a strong height condition that would imply that a group was a cZ-group. That approach failed and was replaced by the notion of computable relative heights. However, I still feel that the idea is interesting and may prove fruitful for some other purpose.     

Characterizing Automorphism Groups of Ordered Abelian Groups

A proof is given of the following theorem, which characterizesfull automorphism groups of ordered abelian groups: a groupH is the automorphism group of some ordered abelian group ifand only if H is right-orderable. 2000 Mathematics Subject Classification20K15, 20K20, 20F60, 20K30 (primary); 03E05 (secondary).     

The bernoulli property for expansive ℤ2 actions on compact groups

We show that an expansive ℤ² action on a compact abelian group is measurably isomorphic to a two-dimensional Bernoulli shift if and only if it has completely positive entropy. The proof uses the algebraic structure of such actions described by Kitchens and Schmidt and an algebraic characterisation of theK property due to Lind, Schmidt and the author. As a corollary, we note that an expansive ℤ² action on a compact abelian group is measurably isomorphic to a Bernoulli shift relative to the Pinsker algebra. A further corollary applies an argument of Lind to show that an expansiveK action of ℤ² on a compact abelian group is exponentially recurrent. Finally an example is given of measurable isomorphism without topological conjugacy for ℤ² actions. Supported in part by N.S.F. grant No. DMS-88-02593 at the University of Maryland and by N.S.F. grant No. DMS-91-03056 at Ohio State University.     

On constructive nilpotent groups

We prove the following: (1) a torsion-free class 2 nilpotent group is constructivizable if and only if it is isomorphic to the extension of some constructive abelian group included in the center of the group by some constructive torsion-free abelian group and some recursive system of factors; (2) a constructivizable torsion-free class 2 nilpotent group whose commutant has finite rank is orderably constructivizable.     

Cubulating small cancellation groups

We study the B(6) and B(4)-T(4) small cancellation groups. These classes include the usual C(1/6) and C(1/4)-T(4) metric small cancellation groups. We show that every finitely presented B(4)-T(4) or word-hyperbolic B(6) group acts properly discontinuously and cocompactly on a CAT(0) cube complex. We show that finitely generated infinite B(6) and B(4)-T(4) groups have codimension 1 subgroups and thus do not have property (T). We show that a finitely presented B(6) group is wordhyperbolic if and only if it contains no subgroup.