ON ELEMENTARY PROPERTIES OF EXISTENTIALLY CLOSED SYSTEMS

has characterized those rings with involution J, with 1, and such that every symmetric element has an inverse under the additional hypothesis that S~~, the subring generated by ,S (the set of symmetric elements), is A, the ring itself.In this paper we raise the question as to when S~ = A under a weaker hypothesis and prove the following theorem.(A henceforth is such that 2A =A and 2x=0 implies x=0 for any xEA.)Theorem 1.Let A be a ring with 1 and suppose that S is a simple Jordan ring under the Jordan multiplication, s o t = st+ts for all s and t in S. Then either S~ = A or SEZ, the center of A, or K, the set o/ skew elements, is an ideal o/ A with K2EZ, K3 = (8), and S is an associative ring under Jordan multiplication.We note that Osborn proves that under his hypothesis, either A is a division ring; the direct sum of two division rings which are antiisomorphic, and / interchanges the summands; or 51 is a field under Jordan multiplication, K is an ideal of A where K2 = (d).A corollary to our result is Corollary 1.Let A be as in Theorem 1. Suppose /urther that A contains no nilpotent ideals.Then either A is simple or A is a direct sum o/ two simple rings which are anti-isomorphic and such that J interchanges the direct summands.

A celebrated theorem of Wedderburn says that a finite associative division ring is a finite (commutative) field. An elegant proof of this, due to E. Artin [3, p. 72], reduces the problem to showing the reduced (or generic) norm has a nontrivial zero, and then applies the theorem (conjectured by Artin and proved by Chevalley) that a polynomial with coefficients in a finite field and zero constant term has a nontrivial zero in that field if its degree is less than the number of variables. The Wedderburn theorem was generalized by A. A. Albert to finite strictly power-associative division rings of characteristic # 2 in [1, p. 301 ] and [2, p. i1 ] (see also [5 ]). The proof reduced the powerassociative case to the associative case by using the properties of Jordan division algebras. It is the purpose of this note to show that Artin's method carries over directly to the power-associative case using the generic norm introduced by N. Jacobson [4]. By avoiding Jordan algebras one is able to extend Albert's result to the characteristic 2 case. The usual definition of a division ring requires that left and right multiplications by a nonzero element be bijective. If the ring is finite it is necessarily an algebra, over a finite field 4, which is algebraic (even finite-dimensional) and without zero divisors. Also, as we shall see later, the algebra necessarily has a unit element. We wish to consider something slightly more general. For our purposes we say an algebraic power-associative algebra is a division algebra if it is unital and no element xO 0 is a zero divisor in the subalgebra (D[x] it generates (since 4<[x] is finite-dimensional, this is equivalent to saying x is invertible in b [x]). Our conditions are less restrictive, since the Jordan algebra of a quadratic form can be a division algebra in our sense with zero divisors. With this definition we have

Commuting maps and identities with inverses on alternative division rings

Several authors have studied the question of when the monoid ring \(DM\) of a monoid \(M\) over a ring \(D\) is a right and/or left fir (free ideal ring), a semifir, or a \(2\)-fir (definitions recalled in §1). It is known that for \(M\) nontrivial, a necessary condition for any of these properties to hold is that \(D\) be a division ring. Under that assumption, necessary and sufficient conditions on \(M\) are known for \(DM\) to be a right or left fir, and various conditions on \(M\) have been proved necessary or sufficient for \(DM\) to be a \(2\)-fir or semifir. A sufficient condition for \(DM\) to be a semifir is that \(M\) be a direct limit of monoids which are free products of free monoids and free groups. Warren Dicks has conjectured that this is also necessary. However F. Cedó has given an example of a monoid \(M\) which is not such a direct limit, but satisfies all the known necessary conditions for \(DM\) to be a semifir. It is an open question whether for this \(M,\) the rings \(DM\) are semifirs. We note here some reformulations of the known necessary conditions for a monoid ring \(DM\) to be a \(2\)-fir or a semifir, motivate Cedó’s construction and a variant thereof, and recover Cedó’s results for both constructions. Any homomorphism from a monoid \(M\) into \(\mathbb {Z}\) induces a \(\mathbb {Z}\)-grading on \(DM,\) and we show that for the two monoids just mentioned, the rings \(DM\) are “homogeneous semifirs” with respect to all such nontrivial \(\mathbb {Z}\)-gradings; i.e., have (roughly) the property that every finitely generated homogeneous one-sided ideal is free of unique rank. If \(M\) is a monoid such that \(DM\) is an \(n\)-fir, and \(N\) a “well-behaved” submonoid of \(M,\) we prove some properties of the ring \(DN.\) Using these, we show that for \(M\) a monoid such that \(DM\) is a \(2\)-fir, mutual commutativity is an equivalence relation on nonidentity elements of \(M,\) and each equivalence class, together with the identity element, is a directed union of infinite cyclic groups or of infinite cyclic monoids. Several open questions are noted.

We are given a division ring D with involution (*) and with a *-valuation V such that V(aa*b − baa*) >0 V(aa*b), for all nonzero elements a, b of D. We assume, further, that the associated residue class division ring D V is a (commutative) field with characteristic 0. In this work, we evidence two criteria in order for D to be either a standard quaternion division algebra or else a purely transcendental extension of its center. We apply one of these to answer our open Question 2.4.3 [2].

Let E*G be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to E*G-isomorphism, there exists at most one Hughes-free division E*G-ring. However, the existence of a Hughes-free division E*G-ring {mathcal {D}}_{E*G} for an arbitrary locally indicable group G is still an open question. Nevertheless, {mathcal {D}}_{E*G} exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether {mathcal {D}}_{E*G} is the universal division ring of fractions in some of these cases. In particular, we show that if G is a residually-(locally indicable and amenable) group, then there exists {mathcal {D}}_{E[G]} and it is universal. In Appendix we give a description of {mathcal {D}}_{E[G]} when G is a RFRS group.

This book provides a general framework for doing geometric group theory for many non-locally-compact topological transformation groups that arise in mathematical practice, including homeomorphism and diffeomorphism groups of manifolds, isometry groups of separable metric spaces and automorphism groups of countable structures. Using Roe's framework of coarse structures and spaces, the author defines a natural coarse geometric structure on all topological groups. This structure is accessible to investigation, especially in the case of Polish groups, and often has an explicit description, generalising well-known structures in familiar cases including finitely generated discrete groups, compactly generated locally compact groups and Banach spaces. In most cases, the coarse geometric structure is metrisable and may even be refined to a canonical quasimetric structure on the group. The book contains many worked examples and sufficient introductory material to be accessible to beginning graduate students. An appendix outlines several open problems in this young and rich theory.

Continuing the work in [ll],[l2] we study division algebras D = k(G) over a field k which are generated by some polycyclic-by-finite subgroup G of the multiplicative group D* of D. We discuss a specific class of examples of such division algebras that can be thought of as multiplicative analogs of the Weyl field. Furthermore, we show that the division algebras D = k(G) always contain free subalgebras of rank ≤ 2, provided G is not abelian-by-finite. Finally, we discuss some open questions concerning commutative subfields and Lie commutators in D = k(G).

In the Eighties G. Cattaneo contributed to Hilbert space quantum mechanics models using the family of splitting subspaces of an inner product space in order to find completeness criteria. In this paper we show how that theory was developed in the last 25 years into a rich theory which has a close connection to quantum information. G. Cattaneo also posed an open problem which was solved only this year.

In this paper we construct a correspondence between a class of irreducible linear groups of finite degree over an associative division ring D and special Jordan rings which are defined by D.

The notion of p-modulus was created, and continues to play a vital role, in complex analysis and geometric function theory. Here we give an overview of the theory of p-modulus on networks that we have been developing in recent years. The hope is not only to develop a flexible tool on networks that can be useful for practical applications, but also that the rich unfolding theory on network will eventually inform the classical theory on metric measure spaces, Euclidean spaces, and the complex plane. We end by offering three open problems that are purely network theory problems. To keep the paper self-contained, we have not included possible applications, both to practical problems and to more theoretic function theoretic problems. We leave those directions for another time.

The von Neumann-Gale Growth Model and its Stochastic Generalization

Symbolic automata and transducers extend finite automata and transducers by allowing transitions to carry predicates and functions over rich alphabet theories, such as linear arithmetic. Therefore, these models extend their classic counterparts to operate over infinite alphabets, such as the set of rational numbers. Due to their expressiveness, symbolic automata and transducers have been used to verify functional programs operating over lists and trees, to prove the correctness of complex implementations of BASE64 and UTF encoders, and to expose data parallelism in computations that may otherwise seem inherently sequential. In this paper, we give an overview of what is currently known about symbolic automata and transducers as well as their variants. We discuss what makes these models different from their finite-alphabet counterparts, what kind of applications symbolic models can enable, and what challenges arise when reasoning about these formalisms. Finally, we present a list of open problems and research directions that relate to both the theory and practice of symbolic automata and transducers.

We study automaton structures, i.e., groups, monoids and semigroups generated by an automaton, which, in this context, means a deterministic finite-state letter-to-letter transducer. Instead of considering only complete automata, we specifically investigate semigroups generated by partial automata. First, we show that the class of semigroups generated by partial automata coincides with the class of semigroups generated by complete automata if and only if the latter class is closed under removing a previously adjoined zero, which is an open problem in (complete) automaton semigroup theory stated by Cain. Then, we show that no semidirect product (and, thus, also no direct product) of an arbitrary semigroup with a (non-trivial) subsemigroup of the free monogenic semigroup is an automaton semigroup. Finally, we concentrate on inverse semigroups generated by invertible but partial automata, which we call automaton-inverse semigroups, and show that any inverse automaton semigroup can be generated by such an automaton (showing that automaton-inverse semigroups and inverse automaton semigroups coincide).

In a recent paper2 we have called a ring 2 primitive if 2 contains a maximal right ideal a such that the quotient (a: W) = (0). The quotient (: 1) is defined to be the largest two-sided ideal of 21 that satisfies the condition 21(a: 21) C a. Thus if 21lias an identity, (a: 21) is the largest two-sided ideal of 21 contained in a. Primitive rings appear to play a role in the general structure theory of rings analogous to that played by simple rings in the Wedderburn-Artin theory. Primitive rings are also fundamental in general representation theory since it is known that a necessary and sufficient condition that a ring 21 be primitive is that 21 be isomorphic to an irreducible ring W of endomorphisms in a commutative group T. If 2 is irreducible in T and Zi is the division ring of endomorphisms commutative with the elements of A, then R is a dense ring of linear transformations in T regarded as a vector space over Z.3 We shall call 2 a left primitive ring if 2 contains a maximal left ideal a' such that the quotient (}': 21)l = (0). Here (s': 2) 1 is the largest two-sided ideal in 2 having the property (a': 21)1 21 C a'. Evidently, a ring is left primitive if and only if it is anti-isomorphic to an irreducible ring of endomorphisms. It is an open question whether or not a primitive ring is necessarily left primitive. This is indeed the case if 2 contains minimal ideals. In many other respects the theory of primitive rings that contain minimal ideals constitutes the most satisfactory part of the theory of primitive rings. An appropriate tool for studying rings of this type is a generalization of the duality theory previously used by Dieudonn6 in studying simple rings that possess minimal ideals.4 In this note we develop this generalization of Dieudonn6's results. We also investigate certain natural topologies that can be defined in any primitive ring.