Countable Field Research Articles

GRÖBNER–SHIRSHOV BASES FOR L-ALGEBRAS

In this paper, we first establish Composition-Diamond lemma for Ω-algebras. We give a Gröbner–Shirshov basis of the free L-algebra as a quotient algebra of a free Ω-algebra, and then the normal form of the free L-algebra is obtained. Second we establish Composition-Diamond lemma for L-algebras. As applications, we give Gröbner–Shirshov bases of the free dialgebra and the free product of two L-algebras, and then we show four embedding theorems of L-algebras: (1) Every countably generated L-algebra can be embedded into a two-generated L-algebra. (2) Every L-algebra can be embedded into a simple L-algebra. (3) Every countably generated L-algebra over a countable field can be embedded into a simple two-generated L-algebra. (4) Three arbitrary L-algebras A, B, C over a field k can be embedded into a simple L-algebra generated by B and C if |k| ≤ dim (B * C) and |A| ≤ |B * C|, where B * C is the free product of B and C.

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.

Jacobson radical non-nil algebras of Gel’fand-Kirillov dimension 2

For an arbitrary countable field, we construct an associative algebra that is graded, generated by finitely many degree-1 elements, is Jacobson radical, is not nil, is prime, is not PI, and has Gel’fand-Kirillov dimension two. This refutes a conjecture incorrectly attributed to Goodearl.

Nil algebras with restricted growth

AbstractIt is shown that over an arbitrary countable field there exists a finitely generated algebra that is nil, infinite dimensional and has Gelfand–Kirillov dimension at most 3.

GRÖBNER–SHIRSHOV BASES AND EMBEDDINGS OF ALGEBRAS

In this paper, by using Gröbner–Shirshov bases, we show that in the following classes, each (respectively, countably generated) algebra can be embedded into a simple (respectively, two-generated) algebra: associative differential algebras, associative Ω-algebras, associative λ-differential algebras. We show that in the following classes, each countably generated algebra over a countable field k can be embedded into a simple two-generated algebra: associative algebras, semigroups, Lie algebras, associative differential algebras, associative Ω-algebras, associative λ-differential algebras. We give another proofs of the well known theorems: each countably generated group (respectively, associative algebra, semigroup, Lie algebra) can be embedded into a two-generated group (respectively, associative algebra, semigroup, Lie algebra).

An exact Ramsey principle for block sequences

We prove an exact, i.e., formulated without $\Delta$-expansions, Ramsey principle for infinite block sequences in vector spaces over countable fields, where the two sides of the dichotomic principle are represented by espectively winning strategies in Gowers' block sequence game and winning strategies in the infinite asymptotic game. This allows us to recover Gowers' dichotomy theorem for block sequences in normed vector spaces by a simple application of the basic determinacy theorem for infinite asymptotic games.

Makar-Limanov's conjecture on free subalgebras

It is proved that over every countable field K there is a nil algebra R such that the algebra obtained from R by extending the field K contains noncommutative free subalgebras of arbitrarily high rank. It is also shown that over every countable field K there is an algebra R without noncommutative free subalgebras of rank two such that the algebra obtained from R by extending the field K contains a noncommutative free subalgebra of rank two. This answers a question of Makar-Limanov [Lenny Makar-Limanov, private communication, Beijing, June 2007].

Stably just infinite rings

We study just infinite algebras which remain so upon extension of scalars by arbitrary field extensions. Such rings are called stably just infinite. We show that just infinite rings over algebraically closed fields are stably just infinite provided that the ring is either right noetherian (4.2) or countably generated over a large field (6.4). We give examples to show that, over countable fields, a just infinite algebra which is either affine or non-noetherian need not remain just infinite under extension of scalars. We also give a concrete classification of PI stably just infinite rings (5.5) and give two characterizations of non-PI stably just infinite rings in terms of Martindale's extended center (3.4), (3.5).

Turing degrees of isomorphism types of algebraic objects

The Turing degree spectrum of a countable structure 𝒜 is the set of all Turing degrees of isomorphic copies of 𝒜. The Turing degree of the isomorphism type of 𝒜, is the least Turing degree in its degree spectrum. We show that there are structures with isomorphism types of arbitrary Turing degrees in each of the following classes: countable fields, rings, and torsion-free Abelian groups of any finite rank. We also show that there are structures in each of these classes the isomorphism types of which do not have Turing degrees. The case of torsion-free Abelian groups of finite rank settles a question left open by Knight, Downey and Jockusch [Downey, Complexity, logic, and recursion theory, Lecture Notes in Pure and Applied Mathematics 187 (ed. A. Sorbi; Marcel Dekker, New York, 1997) 157–205].

Jacobson radical algebras with Gelfand–Kirillov dimension two over countable fields

It is shown that for every countable field K , there is a finitely generated graded Jacobson radical algebra over K of Gelfand–Kirillov dimension two. Examples of finitely generated Jacobson radical algebras of Gelfand–Kirillov dimension two over algebraic extensions of finite fields of characteristic 2 were earlier constructed by Bartholdi [L. Bartholdi, Branch Rings, thinned rings, tree enveloping rings, Israel J. Math. (in press)].

Cardinal invariants of the continuum and combinatorics on uncountable cardinals

We explore the connection between combinatorial principles on uncountable cardinals, like stick and club, on the one hand, and the combinatorics of sets of reals and, in particular, cardinal invariants of the continuum, on the other hand. For example, we prove that additivity of measure implies that Martin’s axiom holds for any Cohen algebra. We construct a model in which club holds, yet the covering number of the null ideal cov ( N ) is large. We show that for uncountable cardinals κ ≤ λ and F ⊆ [ λ ] κ , if all subsets of λ either contain, or are disjoint from, a member of F , then F has size at least cov ( N ) etc. As an application, we solve the Gross space problem under c = ℵ 2 by showing that there is such a space over any countable field. In two appendices, we solve problems of Fuchino, Shelah and Soukup, and of Kraszewski, respectively.

The Classification of Complex Factor-Representations of the Three-Dimensional Heisenberg Group over a Countable Group Field of Finite Characteristic

We consider a field F that is a direct limit of an increasing chain of finite fields, and describe the Bratteli diagram, complex factor-representations, and projective moduli of the Heisenberg group of 3 × 3 upper-triangular matrices with elements from F. Bibliography: 3 titles.

Superdecomposable pure-injective modules exist over some string algebras

We prove that over every non-domestic string algebra over a countable field there exists a superdecomposable pure-injective module.

Gromov translation algebras over discrete trees are exchange rings

It is shown that the Gromov translation ring of a discrete tree over a von Neumann regular ring is an exchange ring. This provides a new source of exchange rings, including, for example, the algebras G(0) of ω x ω matrices (over a field) of constant bandwidth. An extension of these ideas shows that for all real numbers r in the unit interval [0,1], the growth algebras G(r) (introduced by Hannah and O'Meara in 1993) are exchange rings. Consequently, over a countable field, countable-dimensional exchange algebras can take any prescribed bandwidth dimension r in [0,1].

Examples in finite Gel'fand–Kirillov dimension

By modifying constructions of Beı̆dar and Small we prove that for countably generated prime F-algebras of finite GK dimension there exists an affinization having finite GK dimension. Using this result we show: for any field there exists a prime affine algebra of GK dimension two that is neither primitive nor PI; for any countable field F there exists a prime affine F-algebra of GK dimension three that has non-nil Jacobson radical; for any countable field F there exists an affine primitive F-algebra of GK dimension at most four with center equal to a polynomial ring; for a countable field F there exists a primitive affine Jacobson F-algebra of GK dimension three that does not satisfy the Nullstellensatz.

The relation of recursive isomorphism for countable structures

AbstractIt is shown that the relations of recursive isomorphism on countable trees, groups, Boolean algebras, fields and total orderings are universal countable Borel equivalence relations, thus providing a countable analogue of the Borel completeness of the isomorphism relations on these same classes.

A SIMPLE NIL RING EXISTS

ABSTRACT We show that for every countable field K there is a simple nil algebra over K.

Resource-assessment perspectives for unconventional gas systems

Concepts are described for assessing those unconventional gas sys tems that can also be defined as continuous accumulations. Contin uous gas accumulations exist more or less independently of the wa ter column and do not owe their existence directly to the buoyancy of gas in water. They cannot be represented in terms of individual, countable fields or pools delineated by downdip water contacts. For these reasons, traditional resource-assessment methods based on es timating the sizes and numbers of undiscovered discrete fields can not be applied to continuous accumulations. Specialized assessment methods are required. Unconventional gas systems that are also continuous accumu lations include coalbed methane, basin-centered gas, so-called tight gas, fractured shale (and chalk) gas, and gas hydrates. Deep-basin and bacterial gas systems may or may not be continuous accumu lations, depending on their geologic setting. Two basic resource-assessment approaches have been em ployed for continuous accumulations. The first approach is based on estimates of gas in place. A volumetric estimate of total gas in place is commonly coupled with an overall recovery factor to nar row the assessment scope from a treatment of gas volumes residing in sedimentary strata to a prediction of potential additions to re serves. The second approach is based on the production perfor mance of continuous gas reservoirs, as shown empirically by wells and reservoir-simulation models. In these methods, production characteristics (as opposed to gas in place) are the foundation for forecasts of potential additions to reserves.

Amitsur's Conjecture on Polynomial Rings in N Commuting Indeterminates

For each natural number n we construct an algebra over a countable field such that the polynomial ring in n commuting indeterminates over this algebra is Jacobson radical but not a nil-ring. This answers a question of Amitsur.

Classification of Finite Factor Representations of the (2m+1)-Dimensional Heisenberg Group over a Countable Field of Finite Characteristic

For a field F that is the direct limit of an increasing chain of finite fields, we describe the Bratteli diagram, the finite complex factor representations, the Plancherel formula, and the projective modules of the corresponding Heisenberg group.