Uncountable Field Research Articles

Filtered algebraic algebras

Small and Zelmanov posed the question whether every element of a graded algebra over an uncountable field must be nilpotent, provided that the homogeneous elements are nilpotent. This question has recently been answered in the negative by A. Smoktunowicz. In this paper we prove that the answer is affirmative for associated graded algebras of filtered algebraic algebras. Our result is based on Amitsur's theorems on algebas over infinite fields.

A generalization of Amitsur theorem

Suppose that K is a field and R is a K-algebra then R has endomorphism property over K if for each simple right module MR, EndR(M )i s algebraic over K. In this note we will show that; if K is an uncountable field, R is a countably generated K-algebra, M is a countably generated semisimple right R-module and L = EndR(M), then any subring of L containing K is algebraic over K whose restriction to some simple submodule of M is faithful. This extends the well-known theorem of Amitsur.

Periodic groups whose simple modules have finite central endomorphism dimension

Theorem. If k k is an uncountable field and G G is a periodic group with no elements of order the characteristic of k k and if all simple k [ G ] k[G] modules have finite central endomorphism dimension, then G G has an abelian subgroup of finite index.

Sur le groupe des automorphismes polynomiaux du plan affine

Let k be an uncountable field of characteristic 0. We prove that an automorphism of the group of polynomial automorphisms of k 2 is the composition of an interior automorphism and an automorphism of the field k.

Phantom Maps and Purity in Modular Representation Theory, III

In the third installment of this paper, we exhibit two phantom maps between kG-modules, whose composite is not projective, whenever k is an uncountable field of characteristic p and G is a finite group of p-rank at least 2.

Localization and test exponents for tight closure.

We introduce the notion of a test exponent for tight closure, and explore its relationship with the problem of showing that tight closure commutes with localization, a longstanding open question. Roughly speaking, test exponents exist if and only if tight closure commutes with localization: mild conditions on the ring are needed to prove this. We give other, independent, conditions that are necessary and sufficient for tight closure to commute with localization in the general case, in terms of behavior of certain associated primes and behavior of exponents needed to annihilate local cohomology. While certain related conditions (the ones given here are weaker) were previously known to be sufficient, these are the first conditions of this type that are actually equivalent. The difficult calculation of§4 uses associativity of multiplicities and many other tools to show that sufficient conditions for localization to commute with tight closure can be given in which asymptotic statements about lengths of modules defined using the iterates of the Frobenius endomorphism replace the finiteness conditions on sets of primes introduced in §3. The result is local and requires special conditions on the rings: one is that countable prime avoidance holds. This is not a very restrictive condition however: it suffices, for example, for the ring to contain an uncountable field. Countable prime avoidance also holds in any complete local ring. But we also need the existence of a strong test ideal (see

Localisation in rings equipped with a solvable Lie algebra action

Let k be an algebraically closed uncountable field of characteristic 0,g a finite dimensional solvable k-Lie algebraR a noetherian k-algebra on which g acts by k-derivationsU(g) the enveloping algebra of g,A=R*g the crossed product of R by U(g)P a prime ideal of A and Ω(P) the clique of P. Suppose that the prime ideals of the polynomial ring R[x] are completely prime. If R is g-hypernormal, then Ω(P) is classical. Denote by AT the localised ring and let M be a primitive ideal of AT Set Q=P∩R In this note, we show that if R is a strongly (R,g)-admissible integral domain and if QRQ is generated by a regular g-centralising set of elements, then (1)M is generated by a regular g-semi-invariant normalising set of elements of cardinald = dim (RQ 0 + ∣XA (P)∣ (2)d gldim(AT ) = Kdim(AT ) = ht(M) = ht(P).

On free subgroups of units of rings

We prove that if a 2 = b 2 = 0 a^{2}=b^{2}=0 for elements a , b a,b of a ring R R of characteristic zero and a b ab is not nilpotent, then there exists m ∈ N m\in {\mathbf N} such that the group generated by 1 + m a 1+ma and 1 + m b 1+mb is free nonabelian. This is used to prove that a noncommutative positive-definite algebra with involution over an uncountable field contains a free nonabelian subsemigroup.

Gross spaces

A Gross space is a vector space E E of infinite dimension over some field F F , which is endowed with a symmetric bilinear form Φ : E 2 → F \Phi :E^{2} \rightarrow F and has the property that every infinite dimensional subspace U ⊆ E U\subseteq E satisfies dim U ⊥ > U^{\perp }> dim E E . Gross spaces over uncountable fields exist (in certain dimensions) (see [H. Gross and E. Ogg, Quadratic spaces with few isometries, Comment. Math. Helv. 48 (1973), 511-519]). The existence of a Gross space over countable or finite fields (in a fixed dimension not above the continuum) is independent of the axioms of ZFC. Here we continue the investigation of Gross spaces. Among other things, we show that if the cardinal invariant b equals ω 1 \omega _{1} , a Gross space in dimension ω 1 \omega _{1} exists over every infinite field, and that it is consistent that Gross spaces exist over every infinite field but not over any finite field. We also generalize the notion of a Gross space and construct generalized Gross spaces in ZFC.

On a question of Makar-Limanov

Let K K be an uncountable field, let K ⊂ F K \subset F be a field extension, and let A A be an associative K K -algebra. We show that if F ⊗ K A F \otimes _K A contains a non-commutative free algebra, then so does A A .

Properly algebraic and spectrum-finite ideals in Jordan systems

The two main results of this paper are:(i) The set of properly algebraic elements of a Jordan system (algebra, triple system or pair) over an uncountable field is an ideal.(ii) For a semiprimitive Banach Jordan system, the socle, the largest properly algebraic ideal, the largest properly spectrum-finite ideal and the largest von Neumann regular ideal all coincide.

Two P-complete problems in the theory of the reals

A classical subject in computational complexity what usually called algebraic complexity. This area deals with the complexity of algorithms that take their inputs on R n, where R a ring, understood as the number of ring operations the algorithm performs as a function of n and the main kind of results are both upper and lower bounds. A recent survey about this topic [7]. A very special case in algebraic complexity the one dealing with R = ]1%. Here, the assumption that all the elements in the have unit size, and that the cost of the operations also unitary, reflects the particular features of algorithms in numerical analysis. Very recently an article of L. Blum, M. Shub and S. Smale put into this scenery another approach to the complexity: the structural one (see [4]). To do so, they deviced a model of real Turing machines, over the which the basis of a theory of computability and a theory of complexity built. In particular analogs of the classes P and NP over the reals are introduced, and the problem of deciding whether a degree 4 real polynomial has a root shown to be NP-complete, thus, unlikely decidable in polynomial time. On the other hand, for a wide variety of problems, fast parallel algorithms have been designed during the last years. These algorithms use as a model of parallel machine a family of circuits with polynomial size and polylogarithmic depth, for which a logspace uniformity condition valid. However, this is of course not possible for arbitrary constants over an uncountable field (see [6]). For the Boolean model, there are many problems for which algorithms working in polynomial time can be given that solve them and no good fast parallel algorithm known that do the same. Some of these problems share the property that if a fast parallel algorithm found that solves one of them, then every problem in P can be decided by such a fast parallel algorithm. These problems, that are said to be P complete constitute a class whose elements, even when no proof known till now for P ~ NC, are usually considered as problems not admitting a solution in fast parallel time. In section 2 of this paper we propose the parallel real RAM as a model for defining a class of problems dealing with real numbers that are decidable in fast parallel time. This formulation avoids uniformity considerations and the resulting class trivially included in P. We use reductions in this class (in fact, using constant parallel time) to give our main result, the existence of P-complete problems for the real model. In section 3, two

Independence and consistency proofs in quadratic form theory

We consider the following properties of uncountable-dimensional quadratic spaces (E, Φ):(*) For all subspaces U ⊆ E of infinite dimension: dim U˔ < dim E.(**) For all subspaces U ⊆ E of infinite dimension: dim U˔ < ℵ0.Spaces of countable dimension are the orthogonal sum of straight lines and planes, so they cannot have (*), but (**) is trivially satisfied.These properties have been considered first in [G/O] in the process of investigating the orthogonal group of quadratic spaces. It has been shown there (in ZFC) that over arbitrary uncountable fields (**)-spaces of uncountable dimension exist.In [B/G], (**)-spaces of dimension ℵ1 (so (*) = (**)) have been constructed over arbitrary finite or countable fields. But this could be done only under the assumption that the continuum hypothesis (CH) holds in the underlying set theory.

Coherence for polynomial rings

Let II= IIR, be an arbitrary product of copies of R,. We say R is right n-coherent if every finitely generated submodule of Z7 is finitely presented. This notion is called strong coherence in [7], where it seems to have been invented. Here we obtain a new characterization of n-coherence, and study Z7coherence for polynomial rings. Using Goldie’s Theorem, we show that if R is a semiprime two-sided noetherian ring then R[S] is right n-coherent for any set of variables S (Theorem 6). We also obtain the same result for two-sided noetherian rings R, not assumed to be semiprime, provided R contains an uncountable field. Call ring R a left *-ring (star ring) if it has the property that HomA , RR) = ( )* takes finitely generated left R-modules to finitely generated right R modules. The above ideas are connected by the following:

LIE SUPERALGEBRAS WITH BOUNDED DEGREES OF IRREDUCIBLE REPRESENTATIONS

A complete description is given of Lie superalgebras over uncountable fields of characteristic zero with all irreducible representations of finite bounded degree. The even component of such an algebra must be abelian of dimension less than the cardinality of the field, whereas the odd component must contain a submodule of finite codimension with zero multiplication. This condition is also sufficient. Bibliography: 7 titles.

Injective modules, localization and completion in group algebras

Let k be an uncountable field of characteristic zero and G a polycyclic-by-finite group. If P is a prime ideal of kG and L = L(P) the largest Ore set of elements of kG which are regular mod P, it is shown that the injective hull of any simple kG L -module is artinian. The technique used is to construct normal elements in completions of kG at the augmentation ideals of certain abelian normal subgroups. We also show that if N is a normal subgroup of G such that N is torsion free nilpotent of finite rank and G N is polycyclic-by-finite, then the completion ▪ is Noetherian.

On Mori domains and commutative rings with CC ⊥ I

We prove here, among other results, that if A is a Mori domain containing an uncountable field, then any polynomial ring over A is Mori. The Mori property of a domain is determined by its maximal ideals. We use extensively a known characterization of Mori domains in terms of the chain condition on annihilators. We characterize Mori domains in terms of semivaluations and divisibility groups. We prove an analogue of Nagata's theorem for the Mori property and present an example of a completely integrally closed domain A, S a multiplicative subset of A generated by prime elements, such that A is completely integrally closed, but A S is not.

Polynomial Rings Over Goldie Rings are Often Goldie

Here, we prove a result that has as a consequence the fact that if the ring $R$ is an algebra over an uncountable field the a.c.c. on annihilators is preserved under polynomial extensions by any number of variables.

Brauer groups are not characterized by Ulm invariants

Two important invariants of a fieldF are its Brauer groupB(F) and its character groupX(F). IfF is countable, these are countable abelian torsion groups, and so are determined by their Ulm invariants. We show here that Ulm’s invariants do not determine Brauer groups or character groups of uncountable fields. An essential tool, which is entirely group theoretic in nature, is a fact about ultraproducts of torsion groups.

Polynomial rings over Goldie rings are often Goldie

Here, we prove a result that has as a consequence the fact that if the ring R R is an algebra over an uncountable field the a.c.c. on annihilators is preserved under polynomial extensions by any number of variables.