Countable Field Research Articles

Affine Noetherian algebras, filtrations and presentations

Resco and Small gave the first example of an affine Noetherian algebra which is not finitely presented. It is shown that their algebra has no finite-dimensional filtrations whose associated graded algebras are Noetherian, affirming their prediction. A modification of their example yields countable fields over which ‘almost all’ (that is, a co-countable continuum of) affine Noetherian algebras lack such a filtration, and an answer to a question suggested by Irving and Small is derived.

Bilinear spaces over a fixed field are simple unstable

We study the model theory of vector spaces with a bilinear form over a fixed field. For finite fields this can be, and has been, done in the classical framework of full first-order logic. For infinite fields we need different logical frameworks. First we take a category-theoretic approach, which requires very little set-up. We show that linear independence forms a simple unstable independence relation. With some more work we then show that we can also work in the framework of positive logic, which is much more powerful than the category-theoretic approach and much closer to the classical framework of full first-order logic. We fully characterise the existentially closed models of the arising positive theory. Using the independence relation from before we conclude that the theory is simple unstable, in the sense that dividing has local character but there are many distinct types. We also provide positive version of what is commonly known as the Ryll-Nardzewski theorem for ω-categorical theories in full first-order logic, from which we conclude that bilinear spaces over a countable field are ω-categorical.

Real spectra and ℓ-spectra of algebras and vector lattices over countable fields

In an earlier paper we established that every second countable, completely normal spectral space is homeomorphic to the ℓ-spectrum of some Abelian ℓ-group. We extend that result to ℓ-spectra of vector lattices over any countable totally ordered division ring k. Combining those methods with Baro's Normal Triangulation Theorem, we obtain the following result: TheoremFor every countable formally real field k, every second countable, completely normal spectral space is homeomorphic to the real spectrum of some commutative unital k-algebra. The countability assumption on k is necessary: there exists a second countable, completely normal spectral space that cannot be embedded, as a spectral subspace, into either the ℓ-spectrum of any right vector lattice over an uncountable directed partially ordered division ring, or the real spectrum of any commutative unital algebra over an uncountable field. Theorem For every countable formally real field k, every second countable, completely normal spectral space is homeomorphic to the real spectrum of some commutative unital k-algebra.

Enlarging localized polynomial rings while preserving their prime ideal structure

Let n be an integer greater than 1 and let x1,…,xn be indeterminates over a countable field k. In this paper, we employ techniques of Heitmann and Nagata to show there exists an uncountable regular local ring S between the localized polynomial ring k[x1,…,xn](x1,…,xn) and the power series ring k[[x1,…,xn]] such that the prime ideal spectrum of S is homeomorphic to the prime ideal spectrum of k[x1,…,xn](x1,…,xn) as topological spaces with the Zariski topology (Theorem 3.17). Thus S is a local n-dimensional Noetherian domain and the cardinality of the set of prime ideals of S is strictly less than the cardinality of S. We also show that every Noetherian ring A with infinitely many prime ideals has a Noetherian subring B such that the prime ideal spectrum of B is homeomorphic to the prime ideal spectrum of A and the cardinality of the set of prime ideals of B equals the cardinality of B.

A few remarks on invariable generation in infinite groups

A subset [Formula: see text] of a group [Formula: see text] invariably generates [Formula: see text] if [Formula: see text] is generated by [Formula: see text] for any choice of [Formula: see text]. A topological group [Formula: see text] is said to be [Formula: see text] if it is invariably generated by some subset [Formula: see text], and [Formula: see text] if it is topologically invariably generated by some subset [Formula: see text]. In this paper, we study the problem of (topological) invariable generation for linear groups and for automorphism groups of trees. Our main results show that the Lie group [Formula: see text] and the automorphism group of a regular tree are [Formula: see text], and that the groups [Formula: see text] are not [Formula: see text] for countable fields of infinite transcendence degree over a prime field.

Some infinitely generated non-projective modules over path algebras and their extensions under Martin's axiom

In this paper it is proved that, when $Q$ is a quiver that admits some closure, for any algebraically closed field $K$ and any finite dimensional $K$-linear representation $\mathcal{X}$ of $Q$, if ${\rm Ext}^1_{KQ}(\mathcal{X}, KQ) = 0$ then $\mathcal{X}$ is projective. In contrast, we show that if $Q$ is a specific quiver of the type above, then there is an infinitely generated non-projective $KQ$-module $M_{\omega_1}$ such that, when $K$ is a countable field, $\mathbf{MA}_{\aleph_1}$ (Martin's axiom for $\aleph_1$ many dense sets, which is a combinatorial axiom in set theory) implies that ${\rm Ext}^1_{KQ}(M_{\omega_1}, KQ) = 0$.

Invariant universality for quandles and fields

We show that the embeddability relations for countable quandles and for countable fields of any given characteristic other than 2 are maximally complex in a strong sense: they are invariantly universal. This notion from the theory of Borel reducibility states that any analytic quasi-order on a standard Borel space essentially appears as the restriction of the embeddability relation to an isomorphism-invariant Borel set. As an intermediate step we show that the embeddability relation of countable quandles is a complete analytic quasi-order.

On free subsemigroups of associative algebras

In 1992, following earlier conjectures of Lichtman and Makar-Limanov, Klein conjectured that a noncommutative domain must contain a free, multiplicative, noncyclic subsemigroup. He verified the conjecture when the center is uncountable. In this note we consider the existence (or not) of free subsemigroups in associative k k -algebras R R , where k k is a field not algebraic over a finite subfield. We show that R R contains a free noncyclic subsemigroup in the following cases: (1) R R satisfies a polynomial identity and is noncommutative modulo its prime radical. (2) R R has at least one nonartinian primitive subquotient. (3) k k is uncountable and R R is noncommutative modulo its Jacobson radical. In particular, (1) and (2) verify Klein’s conjecture for numerous well-known classes of domains, over countable fields, not covered in the prior literature.

Matrix wreath products of algebras and embedding theorems

We introduce a new construction of matrix wreath products of algebras that is similar to wreath products of groups. We then use it to prove embedding theorems for Jacobson radical, nil, and primitive algebras. In §6, we construct finitely generated nil algebras of arbitrary Gelfand-Kirillov dimension ≥ 8 \geq 8 over a countable field which answers a question from [New trends in noncommutative algebra, Amer. Math. Soc., Providence, RI, 2012, pp. 41–52].

Embedding Theorems for L-Algebras and Anti-commutative Algebras

Bokut, Chen and Huang proved that every countably generated L-algebra over a countable field can be embedded into a simple two-generated L-algebra. In this paper, we prove that every countably generated L-algebra can be embedded into a simple two-generated L-algebra. We also prove that every anti-commutative algebra can be embedded into a simple anti-commutative algebra, and that every countably generated anti-commutative algebra can be embedded into a simple two-generated anti-commutative algebra. Finally, we prove that every anti-commutative algebra can be embedded into its universal enveloping non-associative algebra.

Quasi-normality of idempotents on nilpotents

We study the structure of idempotents in non-Abelian rings, concerning a ring property near to the normality of idempotents on the set of nilpotents. We call a ring with such property right idempotent-quasi-normalizing on nilpotents (simply, right IQNN), and study the structure of right IQNN rings in relation with matrix rings, polynomial ring, and factor rings, by which we extend the class of right IQNN rings. It is proved that the class of IQNN rings contains the 2 by 2 full matrix rings over fields and the upper triangular matrix rings over reduced rings. It is shown that given any countable field K, there exists a semiprime IQNN algebra R over K such that the polynomial ring R[x] over R is IQNN but not NI, and the upper nilradical of R[x] is zero.

A COMPUTABLE FUNCTOR FROM GRAPHS TO FIELDS

AbstractFried and Kollár constructed a fully faithful functor from the category of graphs to the category of fields. We give a new construction of such a functor and use it to resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure${\cal S}$, there exists a countable field${\cal F}$of arbitrary characteristic with the same essential computable-model-theoretic properties as${\cal S}$. Along the way, we develop a new “computable category theory”, and prove that our functor and its partially defined inverse (restricted to the categories of countable graphs and countable fields) are computable functors.

On the topology of valuation-theoretic representations of integrally closed domains

Let F be a field. For each nonempty subset X of the Zariski–Riemann space of valuation rings of F, let A(X)=⋂V∈XV and J(X)=⋂V∈XMV, where MV denotes the maximal ideal of V. We examine connections between topological features of X and the algebraic structure of the ring A(X). We show that if J(X)≠0 and A(X) is a completely integrally closed local ring that is not a valuation ring of F, then there is a space Y of valuation rings of F that is perfect in the patch topology such that A(X)=A(Y). If any countable subset of points is removed from Y, then the resulting set remains a representation of A(X). Additionally, if F is a countable field, the set Y can be chosen homeomorphic to the Cantor set. We apply these results to study properties of the ring A(X) with specific focus on topological conditions that guarantee A(X) is a Prüfer domain, a feature that is reflected in the Zariski–Riemann space when viewed as a locally ringed space. We also classify the rings A(X) where X has finitely many patch limit points, thus giving a topological generalization of the class of Krull domains, one that includes interesting Prüfer domains. To illustrate the latter, we show how an intersection of valuation rings arising naturally in the study of local quadratic transformations of a regular local ring can be described using these techniques.

Infinite dimensional representations of finite dimensional algebras and amenability

We present a novel quantitative approach to the representation theory of finite dimensional algebras motivated by the emerging theory of graph limits. We introduce the rank spectrum of a finite dimensional algebra R over a countable field. The elements of the rank spectrum are representations of the algebra into von Neumann regular rank algebras, and two representations are considered to be equivalent if they induce the same Sylvester rank functions on R-matrices. Based on this approach, we can divide the finite dimensional algebras into three types: finite, amenable and non-amenable representation types. We prove that string algebras are of amenable representation type, but the wild Kronecker algebras are not. Here, the amenability of the rank algebras associated to the limit points in the rank spectrum plays a very important part. We also show that the limit points of finite dimensional representations of algebras of amenable representation type can always be viewed as representations of the algebra in the continuous ring invented by John von Neumann in the 1930’s. As an application in algorithm theory, we introduce and study the notion of parameter testing of modules over finite dimensional algebras, that is analogous to the testing of bounded degree graphs introduced by Goldreich and Ron. We shall see that for string algebras all the reasonable (stable) parameters are testable.

Measure preserving actions of affine semigroups and patterns

Ergodic and combinatorial results obtained in Bergelson and Moreira [Ergodic theorem involving additive and multiplicative groups of a field and$\{x+y,xy\}$patterns.Ergod. Th. & Dynam. Sys.to appear, published online 6 October 2015, doi:10.1017/etds.2015.68], involved measure preserving actions of the affine group of a countable field$K$. In this paper, we develop a new approach, based on ultrafilter limits, which allows one to refine and extend the results obtained in Bergelson and Moreira,op. cit., to a more general situation involving measure preserving actions of thenon-amenableaffine semigroups of a large class of integral domains. (The results and methods in Bergelson and Moreira,op. cit., heavily depend on the amenability of the affine group of a field.) Among other things, we obtain, as a corollary of an ultrafilter ergodic theorem, the following result. Let$K$be a number field and let${\mathcal{O}}_{K}$be the ring of integers of$K$. For any finite partition$K=C_{1}\cup \cdots \cup C_{r}$, there exists$i\in \{1,\ldots ,r\}$such that, for many$x\in K$and many$y\in {\mathcal{O}}_{K}$,$\{x+y,xy\}\subset C_{i}$.

Polynomial recurrence with large intersection over countable fields

We give a short proof of polynomial recurrence with large intersection for additive actions of finite-dimensional vector spaces over countable fields on probability spaces, improving upon the known size and structure of the set of strong recurrence times.

Ramsey algebras and the existence of idempotent ultrafilters

Hindman's Theorem says that every finite coloring of the positive natural numbers has a monochromatic set of finite sums. Ramsey algebras, recently introduced, are structures that satisfy an analogue of Hindman's Theorem. It is an open problem posed by Carlson whether every Ramsey algebra has an idempotent ultrafilter. This paper developes a general framework to study idempotent ultrafilters. Under certain countable setting, the main result roughly says that every nondegenerate Ramsey algebra has a nonprincipal idempotent ultrafilter in some nontrivial countable field of sets. This amounts to a positive result that addresses Carlson's question in some way.

Ergodic theorem involving additive and multiplicative groups of a field and patterns

We establish a ‘diagonal’ ergodic theorem involving the additive and multiplicative groups of a countable field $K$ and, with the help of a new variant of Furstenberg’s correspondence principle, prove that any ‘large’ set in $K$ contains many configurations of the form $\{x+y,xy\}$. We also show that for any finite coloring of $K$ there are many $x,y\in K$ such that $x,x+y$ and $xy$ have the same color. Finally, by utilizing a finitistic version of our main ergodic theorem, we obtain combinatorial results pertaining to finite fields. In particular, we obtain an alternative proof for a result obtained by Cilleruelo [Combinatorial problems in finite fields and Sidon sets. Combinatorica32(5) (2012), 497–511], showing that for any finite field $F$ and any subsets $E_{1},E_{2}\subset F$ with $|E_{1}|\,|E_{2}|>6|F|$, there exist $u,v\in F$ such that $u+v\in E_{1}$ and $uv\in E_{2}$.

D sets and a Sárközy theorem for countable fields

We establish that if ℱ is a countable field of characteristic p, U is a unitary action of ℱl on a Hilbert space ℋ, and P is an essential idempotent ultrafilter on ℱ, then for every polynomial r: ℱ → ℱl with r(0) = 0 one has $$P - {\lim _g}{U_{r(g)}}f = {P_r}f{\rm{ weakly,}}$$ , where Pr is the projection onto the closed subspace $${H_r} = \{ f \in H:{\{ {U_{r(g)}}f\} _{g \in F}}{\text{ is precompact in the strong topology\} }}{\text{.}}$$ . We then derive combinatorial consequences of this result, including results for sufficiently large finite fields.

Simplifying Smoktunowicz's Extraordinary Example

At the turn of the 21st century Agata Smoktunowicz constructed the first example of a nil algebra over a countable field such that the polynomial ring over the algebra is not nil. This answered an old question of Amitsur. We present a simplification of the example.