Galois stratification overe-fold ordered Frobenius fields

Ane-fold ordered field is said to be Frobenius if it is a PRCe field which has the embedding property. By means of a Galois stratification procedure we prove that the theory ofe-fold ordered Frobenius fields is decidable. Research conducted at the Max-Planck-Institut für Mathematik, Bonn. This work constitutes a part of the Ph.D. dissertation done at the K.U. Leuven under the supervision of Prof. Moshe Jarden and Prof. Jan Denef.     

Stochastic games with two non-absorbing states

In the present paper we consider recursive games that satisfy an absorbing property defined by Vieille. We give two sufficient conditions for existence of an equilibrium payoff in such games, and prove that if the game has at most two non-absorbing states, then at least one of the conditions is satisfied. Using a reduction of Vieille, we conclude that every stochastic game which has at most two non-absorbing states admits an equilibrium payoff. This paper is part of the Ph.D. thesis of the author completed under the supervision of Prof. Abraham Neyman at the Hebrew University of Jerusalem. I would like to thank Prof. Neyman for many discussions and ideas and for the continuous help he offered. I also thank Nicolas Vieille for his comments on earlier versions of the paper.     

Independent sets in regular graphs

Lower and upper bounds for the maximal number of independent vertices in a regular graph are obtained, it is shown that the bounds are best possible. Some properties of regular graphs concerning the property ℋ defined below are investigated. This paper is to be a part of the author’s Ph. D. thesis written under the supervision of Prof. B. Grünbaum at the Hebrew University of Jerusalem.     

A remark on almost extensions of lipschitz functions

We present a simpler proof of a result of J. Bourgain on almost extensions of functions satisfying a Lipschitz condition onδ-nets. This is part of the author’s Master Thesis written at the Technion — Israel Institute of Technology, under the supervision of Prof. Yoav Benyamini.     

Structure of quasi-invariant sets

The content of this paper corresponds to a part of the author's Ph.D. thesis research carried out in Tel-Aviv University under the supervision of Prof. G. Freiman and Prof. M. Herzog.     

The product of all the irreducible characters of a finite simple group

The content of this paper corresponds to a part of the author's Ph.D. thesis carried out in Tel Aviv University under the supervision of Prof. Marcel Herzog.     

On Fragility of Generalizations of Factoriality

Any class of domains, in particular a class of domains that arises from generalizations of factoriality, invites questions about its stability under the standard operations. One of these generalizations of factoriality is the one that requires that every nonzero element be contained in only finitely many principal prime ideals of height one. We use this property to settle all the open cases in the literature on stability of generalizations of factoriality under the standard ring extensions. The paper provides a compendium on the stability, under ring extensions, of all the known generalizations of factoriality. We also use stability properties of factorization in extensions of valuation domains to give a new characterization of discrete valuation domains.     

The one-block property in varieties of semigroups

The one-block property is stated and it is proved that, whenever it holds for a class closed under homomorphic images, it implies congruence semimodularity for the whole class. An equational characterization of regular varieties having the one-block property is obtained. Some characterization is obtained also for irregular varieties of semigroups having the one-block property. This paper constitutes part of a dissertation submitted to the Graduate Division of the University of Hawaii in partial fulfillment of the requirement for the PhD in Mathematics. The dissertation was written under the supervision of Prof. J.B. Nation     

Undecidability of pseudop-adically closed fields

The content of this paper corresponds to a part of the author's Ph.D. thesis carried out in Tel Aviv University under the supervision of Prof. Moshe Jarden. The work was partially supported by a grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development.     

Prime representations from a homological perspective

We explore the relation between self extensions of simple representations of quantum affine algebras and the property of a simple representation being prime. We show that every nontrivial simple representation has a nontrivial self extension. Conversely, we prove that if a simple representation has a unique nontrivial self extension up to isomorphism, then its Drinfeld polynomial is a power of the Drinfeld polynomial of a prime representation. It turns out that, in the $\mathfrak{sl }_2$ -case, a simple module is prime if and only if it has a unique nontrivial self extension up to isomorphism. It is tempting to conjecture that this is true in general and we present a large class of prime representations satisfying this homological property.     

Some extensions of weakly mixing flows

Using a technique of R. Ellis we prove the existence of many weakly mixing (w.m.) flows which are distal extensions of a given w.m. flow. Then we indicate two w.m. minimal flows whose product has a minimal non-w.m. subflow. This work was done under the helpful supervision of Professor H. Furstenberg as a part of the author’s Ph.D. thesis to be submitted to the Hebrew University of Jerusalem.     

Quantum groups of dimensionpq 2

In this paper we construct two families of non-trivial self-dual semisimple Hopf algebras of dimensionpq ² and investigate closely their (quasi) triangular structures. The paper contains also general results on finite-dimensional triangular Hopf algebras, unimodularity, semisimplicity and ribbon structures of finite-dimensional semisimple Hopf algebras. Dedicated to the memory of Professor S. A. Amitsur This work was partially supported by the Basic Research Foundation administrated by the Israel Academy of Sciences and Humanities. The paper consists of part of the author’s doctoral dissertation written at Ben Gurion University under the supervision of Prof. M. Cohen and Prof. D. E. Radford whom the author wishes to thank for their valuable guidance.     

Extensions of commutative rings

In studying the minimal prime spectra of commutative rings with identity we have been able to identify several interesting types of extensions of rings. In particular, we determine what kind of ring extensions will result in a homeomorphisms of the hull-kernel and inverse topologies on the minimal prime spectra. We relate these types of extensions to other known types of extensions.     

Projectivity,prime ideals,and chain conditions of Stone algebras

Some properties of projective stone algebras are exhibited, which are connected with the ordered set of prime ideals. From this we derive a simple characterization of finite projective Stone algebras, and of those projective Stone algebras, whose centre is a projective Boolean algebra, and whose dense set is a projective Stone algebras, whose centre is a projective Boolean algebra, and whose dense set is a projective distributive lattice. Finally, we give some conditions under which a Stone algebra has no chains of type λ, where λ is an infinite regular cardinal. The results of this paper are part of the author's Ph.D. Thesis written under the direction of S. Koppelberg. The author wishes to express his gratitude to Prof. Koppelberg for her guidance and her patience. Presented by K. A. Baker.     

On the model companion fore-foldp-adically valued fields

We describe a model companion for the first-order theory ofe-foldp-adically valued fields. We obtain several characterizations of this model companion by means of geometro-algebraic, Galois-theoretic and measure-theoretic properties. This note corresponds to a chapter in the author's Ph.D. thesis, carried out in Tel-Aviv University under the supervision of Prof. Moshe Jarden. The work was partially supported by a grant from the G.I.F., the German-Israeli foundation for scientific research and development.     

Finite Normalizing Extensions of FSN Rings

Fully semiprimary Noetherian bimodules and their bimodule extensions are examined. In the presence of incomparability of the link graph of prime ideals, certain bimodule extensions preserve the fully semiprimary property. In particular, a finite normalizing extension ring of a fully semiprimary Noetherian ring is also fully semiprimary as a bimodule over the base ring. It is shown that the extension ring is itself a fully semiprimary ring. An application to crossed products over finite groups is given.     

Prime Ideals in Birational Extensions of Two-Dimensional Power Series Rings

In this article we consider simple birational extensions of power series rings in one variable over one-dimensional Noetherian domains having infinitely many maximal ideals. For these rings we describe the partially ordered sets that arise as prime spectra. We characterize the prime spectra in the case that the coefficient rings are countable Dedekind domains. The prime spectra over Dedekind domains are the same as the prime spectra that arise for simple birational extensions of power series rings over the integers and the same as the prime spectra of simple birational extensions of k[[x]][z], where k is a countable field and x and z are indeterminates.     

Proprietes et applications de la notion de contenu

The notion of content is used to solve certain problems. In the first part, we show that the structural morphism of a content algebra (see the paper of D.E. Rush [22] ) is spectrally open, under mild hypothesis. We show also that a flat module is universally content if and only if it is a Mittag-Leffler’s module in the sense of f2ll . In the second part, using content, we exhibit a kind of localization of a commutative ring A, attached to eyery subset X of Spec(A) i.e. a flat morphism A→ X(A). We can thus show that every quasi-compact, stable under generization subset of a spectra is a spectral image under a flat morphism, in a canonical way. We can also give in certain cases an elementary construction of the maximal flat injective epimor-phism of a ring. Suppose that A is a Noetherian ring and consider pro-perties of Noetherian rings such as factoriality, normality and so on.

Let X be the set of prime ideals of A at which A has the property. If X is stable under generization, the flat morphism A→ X(A) verifies hin general the ring X(A) has lornlly the property and a prime ideal P of A has a prime ideal lying over in X(A) if and only if pthe ring has the property at P.     

Thick subcategories of modules over commutative noetherian rings (with an appendix by Srikanth Iyengar)

For a commutative noetherian ring A, we compare the support of a complex of A-modules with the support of its cohomology. This leads to a classification of all full subcategories of A-modules which are thick (that is, closed under taking kernels, cokernels, and extensions) and closed under taking arbitrary direct sums. In addition, subcategories of A-modules that are closed under taking submodules, extensions, and direct unions are classified via associated prime ideals.     

Pro- $$\mathcal {C}$$ C congruence properties for groups of rooted tree automorphisms

We propose a generalisation of the congruence subgroup problem for groups acting on rooted trees. Instead of only comparing the profinite completion to that given by level stabilizers, we also compare pro- $$\mathcal {C}$$ completions of the group, where $$\mathcal {C}$$ is a pseudo-variety of finite groups. A group acting on a rooted, locally finite tree has the $$\mathcal {C}$$ -congruence subgroup property ( $$\mathcal {C}$$ -CSP) if its pro- $$\mathcal {C}$$ completion coincides with the completion with respect to level stabilizers. We give a sufficient condition for a weakly regular branch group to have the $$\mathcal {C}$$ -CSP. In the case where $$\mathcal {C}$$ is also closed under extensions (for instance the class of all finite p-groups for some prime p), our sufficient condition is also necessary. We apply the criterion to show that the Basilica group and the GGS-groups with constant defining vector (odd prime relatives of the Basilica group) have the p-CSP.