The Structure of the Tame Kernels of Quadratic Number Fields (III)

Let F be an imaginary quadratic number field and K ₂ O _F the tame kernel of F. In this article, we determine all possible values of r ₄(K ₂ O _F ) for each type of imaginary quadratic number field F. In particular, for each type of imaginary quadratic number field we give the maximum possible value of r ₄(K ₂ O _F ) and show that each integer between the lower and upper bounds occurs as a value of the 4-rank of K ₂ O _F for infinitely many imaginary quadratic number fields F.     

APPLICATIONS OF CLIFFORD ALGEBRAS TO INVOLUTIONS AND QUADRATIC FORMS

ABSTRACT

Let k be a field, char k ≠ 2, F = k(x), D a biquaternion division algebra over k, and σ an orthogonal involution on D with nontrivial discriminant. We show that there exists a quadratic form ? ∈ I ²(F) such that dim ? = 8, [C(?)] = [D], and ? does not decompose into a direct sum of two forms similar to two-fold Pfister forms. This implies in particular that the field extension F(D)/F is not excellent. Also we prove that if A is a central simple K-algebra of degree 8 with an orthogonal involution σ, then σ is hyperbolic if and only if σ _K(A) is hyperbolic. Finally, let σ be a decomposable orthogonal involution on the algebra M _{2 ^m} (K). In the case m ≤ 5 we give another proof of the fact that σ is a Pfister involution. If m ≥ 2 ^n?2 ? 2 and n ≥ 5, we show that q _σ ∈ I ⁿ (K), where q _σ is a quadratic form corresponding to σ. The last statement is founded on a deep result of Orlov et al. (2000) concerning generic splittings of quadratic forms.     

On the rank of the 2-class group of an extension of degree 8 over $$\mathbb {Q}$$ Q

Let K be an imaginary cyclic quartic number field whose 2-class group is nontrivial, it is known that there exists at least one unramified quadratic extension F of K. In this paper, we compute the rank of the 2-class group of the field F.

     

The Small Index Property for Free Nilpotent Groups

Let F be a relatively free algebra of infinite rank ?. We say that F has the small index property if any subgroup of Γ = Aut(F) of index at most ? contains the pointwise stabilizer Γ_(U) of a subset U of F of cardinality less than ?. We prove that every infinitely generated free nilpotent/abelian group has the small index property, and discuss a number of applications.     

On the 2-class field tower of some imaginary biquadratic number fields

Let be an imaginary biquadratic number field with Cl_k,2, the 2-class group of k, isomorphic to Z/2Z × Z/2^mZ, m > 1, with q a prime congruent to 3 mod 4 and d a square-free positive integer relatively prime to q. For a number of fields k of the above type we determine if the 2-class field tower of k has length greater than or equal to 2. To establish these results we utilize capitulation of ideal classes in the three unramified quadratic extensions of k, ambiguous class number formulas, results concerning the fundamental units of real biquadratic number fields, and criteria for imaginary quadratic number fields to have 2-class field tower length 1. 2000 Mathematics Subject Classification Primary—11R29     

The Cohen–Macaulay and Gorenstein Properties of Rings Associated to Filtrations

Let (R, m) be a Cohen–Macaulay local ring, and let ? = {F _i } _i∈? be an F ₁-good filtration of ideals in R. If F ₁ is m-primary we obtain sufficient conditions in order that the associated graded ring G(?) be Cohen–Macaulay. In the case where R is Gorenstein, we use the Cohen–Macaulay result to establish necessary and sufficient conditions for G(?) to be Gorenstein. We apply this result to the integral closure filtration ? associated to a monomial parameter ideal of a polynomial ring to give necessary and sufficient conditions for G(?) to be Gorenstein. Let (R, m) be a Gorenstein local ring, and let F ₁ be an ideal with ht(F ₁) = g > 0. If there exists a reduction J of ? with μ(J) = g and reduction number u: = r _J (?), we prove that the extended Rees algebra R′(?) is quasi-Gorenstein with a-invariant b if and only if J ⁿ : F _u  = F _n+b?u+g?1 for every n ∈ ?. Furthermore, if G(?) is Cohen–Macaulay, then the maximal degree of a homogeneous minimal generator of the canonical module ω _G(?) is at most g and that of the canonical module ω _R′(?) is at most g ? 1; moreover, R′(?) is Gorenstein if and only if J ^u : F _u  = F _u . We illustrate with various examples cases where G(?) is or is not Gorenstein.     

Range-Inclusive Maps in Rings with Idempotents

Let 𝒜 be a ring, let ? be an 𝒜-bimodule, and let 𝒞 be the center of ?. A map F:𝒜 → ? is said to be range-inclusive if [F(x), 𝒜] ? [x, ?] for every x ∈ 𝒜. We show that if 𝒜 contains idempotents satisfying certain technical conditions (which we call wide idempotents), then every range-inclusive additive map F:𝒜 → ? is of the form F(x) = λx + μ(x) for some λ ∈ 𝒞 and μ:𝒜 → 𝒞. As a corollary we show that if 𝒜 is a prime ring containing an idempotent different from 0 and 1, then every range-inclusive additive map from 𝒜 into itself is commuting (i.e., [F(x), x] = 0 for every x ∈ 𝒜).     

On 2-von Neumann Regular Rings #

In this article, we consider 2-von Neumann regular rings, that is, rings R with the property that, if F ₂ → F ₁ → F ₀ → E → 0 is an exact sequence of R-modules with F ₀, F ₁, and F ₂ finitely generated free modules, then the module E is projective. For each positive integer m, as well as for m = ∞, we exhibit a class of 2-von Neumann regular rings with Krull dimension m. For this purpose, we study trivial extensions of local rings by infinite-dimensional vector spaces over their residue fields. The article includes a brief discussion of the scope and precision of our results.     

Metrizability of Spaces of ℝ-places of Function Fields of Transcendence Degree 1 Over Real Closed Fields

In this article we discuss the following question “When do different orderings of the rational function field R(X) (where R is a real closed field) induce the same ?-place?” We use this to show that if R contains a dense real closed subfield R′, then the spaces of ?-places of R(X) and R′(X) are homeomorphic. For the function field K = R(X), we prove that its space M(K) of ?-places is metrizible if and only if R contains a countable dense subfield. Moreover, we show that this condition is neccessary for the metrizability of M(F) for any function field F of transcendence degree 1 over R.     

A GENERALIZATION OF AUSLANDER–BUCHSBAUM AND BASS FORMULAS

Abstract

Given a contravariant functor F : 𝒞 → 𝒮ets for some category 𝒞, we say that F (𝒞) (or F) is generated by a pair (X, x) where X is an object of 𝒞 and x ∈ F(X) if for any object Y of 𝒞 and any y ∈ F(Y), there is a morphism f : Y → X such that F(f)(x) = y. Furthermore, when Y = X and y = x, any f : X → X such that F(f)(x) = x is an automorphism of X, we say that F is minimally generated by (X, x). This paper shows that if the ring R is left noetherian, then there exists a minimal generator for the functor ?xt (?, M) : ? → 𝒮ets, where M is a left R-module and ? is the class (considered as full subcategory of left R-modules) of injective left R-modules.     

On Diophantine equation 3a 2 x 4 − By 2 = 1

Bumby proved that the only positive solutions to the quartic Diophantine equation 3x ⁴ ? 2y ² = 1 are (x, y) = (1, 1), (3, 11). In this paper, we extend this result and prove that if the class number of the field ${{\rm Q}(\sqrt{1-3a^{2}})}Bumby proved that the only positive solutions to the quartic Diophantine equation 3x ⁴ − 2y ² = 1 are (x, y) = (1, 1), (3, 11). In this paper, we extend this result and prove that if the class number of the field Q(?{1-3a²}){{\rm Q}(\sqrt{1-3a^{2}})} is not divisible by 2, the equation 3a ² x ⁴ − By ² = 1 has at most two solutions. However, both solutions occur in only one case, a = 1, b = 2, as solved by Bumby. The proof utilizes the law of quadratic reciprocity that seems very rare in solving Diophantine equations, and the solution will be also obtained effectively through the proof when it exists.     

ON K 2 OF DIVISION ALGEBRAS

ABSTRACT

In this paper, it is proved that if F is a global field, then for any integer n > 3, there is an extension field E over F of degree n such that K ₂ E is not generated by the Steinberg symbols {a, b} with a ∈ F*, b, ∈ E*. If however, F is a number field and D is a finite-dimensional central division F-algebra with square free index, then K ₂ D is always generated by the Steinberg symbols {a, b} with a ∈ F*, b ∈ D*. Finally, the tame kernels of central division algebras over F are expressed explicitly.     

Application of Greenberg's conjecture in the capitulation problem

Let r be a positive integer. Assume Greenberg's conjecture for some totally real number fields, we show that there exists an infinite family of imaginary cyclic number fields F over the field of rational number field , with an elementary 2‐class group of rank equal to r that capitulates in an unramified quadratic extension over F. Also, we give necessary and sufficient conditions for the Galois group of the unramified maximal 2‐extension over F to be abelian.     

A Criterion for the Triviality of G(D) and Its Applications to the Multiplicative Structure of D

Let D be an F-central division algebra of index n. Here we present a criterion for the triviality of the group G(D) = D*/Nrd _D/F (D*)D′ and thus generalizing various related results published recently. To be more precise, it is shown that G(D) = 1 if and only if SK ₁(D) = 1 and F ^*2 = F ^*2n . Using this, we investigate the role of some particular subgroups of D* in the algebraic structure of D. In this direction, it is proved that a division algebra D of prime index is a symbol algebra if and only if D* contains a non-abelian nilpotent subgroup. More applications of this criterion including the computation of G(D) and the structure of maximal subgroups of D* are also investigated     

Commuting Graphs of Matrix Algebras

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R, and two distinct vertices x and y are adjacent if and only if xy = yx. The commuting graph of a group G, denoted by Γ(G), is similarly defined. In this article we investigate some graph-theoretic properties of Γ(M _n (F)), where F is a field and n ≥ 2. Also we study the commuting graphs of some classical groups such as GL _n (F) and SL _n (F). We show that Γ(M _n (F)) is a connected graph if and only if every field extension of F of degree n contains a proper intermediate field. We prove that apart from finitely many fields, a similar result is true for Γ(GL _n (F)) and Γ(SL _n (F)). Also we show that for two fields F and E and integers n, m ≥ 2, if Γ(M _n (F))?Γ(M _m (E)), then n = m and |F|=|E|.     

Group Laws [x,y −1] ≡ u(x,y) and Varietal Properties

Let F = ?x, y? be a free group. It is known that the commutator [x, y ^?1] cannot be expressed in terms of basic commutators, in particular in terms of Engel commutators. We show that the laws imposing such an expression define specific varietal properties. For a property 𝒫 we consider a subset U(𝒫) ? F such that every law of the form [x, y ^?1] ≡ u, u ∈ U(𝒫) provides the varietal property 𝒫. For example, we show that each subnormal subgroup is normal in every group of a variety 𝔙 if and only if 𝔙 satisfies a law of the form [x, y ^?1] ≡ u, where u ∈ [F′, ?x?].     

Solvability of a Commutative Algebra which Satisfies (x 2)2 = 0

We studied the solvability of the algebra which satisfies the polynomial identity (x ²)² = 0. We believe that, if A is a finite dimensional commutative algebra over a field F of characteristic not 2 which satisfies (x ²)² = 0 for all x ∈ A, then A is solvable. In this article we proved this when dim  _F A ≤ 7.     

Star-Configurations in ℙ2 Having Generic Hilbert Function and the Weak Lefschetz Property

We prove that a star-configuration 𝕏 in ?² is defined by general forms of degrees ≤2 if and only if 𝕏 has generic Hilbert function. We also show that if 𝕏 and 𝕐 are star-configurations in ?² defined by general forms of degrees ≤2 and σ(𝕏) ≠ σ(𝕐), then the ring R/(I _𝕏 + I _𝕐) has the Weak Lefschetz property. These two results generalize results of Ahn and Shin [3 Ahn , J. , Shin , Y. S. ( 2012 ). The minimal free resolution of a star-configuration in ? ⁿ and the weak lefschetz property . J. Korean Math. Soc. 49 ( 2 ): 405 – 417 . [Google Scholar]]. Furthermore, we find the Lefschetz element of the graded Artinian ring R/(I _𝕏 + I _?) precisely when 𝕏 and ? are two star-configurations in ?² defined by general forms F ₁,…, F _s , and G ₁,…, G _s , L, respectively, with deg F _i  = deg G _i  = 2 for every i ≥ 1, and deg L = 1 with s ≥ 3.     

A Note on the Strong Containment for Locally Defined Formations

Let 𝔉 =LF(F) and ? =LF(H) be two locally defined formations such that 𝔉 ? ?. In this article, we will find necessary conditions to have H(p) ? f*(p) for a fixed prime number p.     

A Note on Range-Inclusive Maps in Rings with Idempotents

Let A be a ring, let M be an A-bimodule, and let C be the center of M. A map F: A → M is said to be range-inclusive if [F(x), A] ? [x, M] for every x ? A. Recently, Bre?ar proved that if A is a unital ring and M a unital A-bimodule such that A contains wide idempotents, then every range-inclusive additive map F: A → M is of the form F(x) = λx + μ(x) for some λ ?C and μ: A → C. Our main purpose is to remove the assumption of unitality in the above result.