Dedekind Ring Research Articles

Separable torsion-free modules with UA-rings of endomorphisms

In this paper we study a property of unique addition for an endomorphisms ring of torsion-free separable module over commutative Dedekind ring.

A Norm Principle for Class Groups of Reductive Group Schemes over Dedekind Rings of Integers of Local and Global Fields

We discuss and prove some results on Corestriction principle for non-abelian etale cohomology and Norm principle for class groups of reductive group schemes over Dedekind rings in global fields.

On Finite Nilpotent Matrix Groups over Integral Domains

We consider finite nilpotent groups of matrices over commutative rings. A general result concerning the diagonalization of matrix groups in the terms of simple conditions for matrix entries is proven. We also give some arithmetic applications for representations over Dedekind rings.

On the distribution of eigenspaces in classical groups over finite rings

Jeffrey D. Achter establishes in [2] a connection between the distribution of class groups of function fields and the distribution of eigenspaces in symplectic groups. Gunter Malle relates this idea in [10] to the number field case. Motivated by these texts we compute here for a finite p-torsion R-module H, where R is a Dedekind ring with finite quotients and q=|R/p|, the limitPG,∞,q,f(H):=limn→∞PG,n,q,f(H):=limn→∞|{g∈Gn(R/pf)|ker(g−1)≅H}||Gn(R/pf)| for certain classical groups G of increasing dimension n. In doing so we extend the results of Jason Fulman ([4] and [5]) concerning distributions of eigenspaces over finite fields. Furthermore we give a reasonable backup for the conjecture of G. Malle (2.1 in [10]).

Locally soluble AFA-groups

Let A be an R G-module, where R is a ring, G is a locally soluble group, C G (A) = 1; and every proper subgroup H of G for which A = C A (H) is not an Artinian R-module is finitely generated. It is shown that a locally soluble group G satisfying these conditions is hyper-Abelian if R is a Dedekind ring. We describe the structure of the group G in the case where G is a finitely generated soluble group, A = C A (G) is not an Artinian R-module, and R is a Dedekind ring.

UNITRIANGULAR FACTORIZATION OF TWISTED CHEVALLEY GROUPS

Unitriangular factorization is a presentation of a linear group as a product of unipotent radicals of a Borel subgroup and its opposite. Whether this decomposition is known for Chevalley groups over rings of stable rank 1 and some Dedekind rings of arithmetic type, the case of twisted groups has been studied only over finite fields. In the present paper we give a much simpler proof for twisted groups over finite fields and the field of complex numbers.

Steinitz classes for Galois extensions of Dedekind rings

Galois extensions of commutative rings have been studied by Chase, Harrison, Rosenberg and others. Suppose B|A is such an extension with (finite) group G where both A and B are Dedekind rings. Then the Steinitz class s B|A is an element in the class group Cl(A) which vanishes if and only if B is a free A-module. It is shown that s B|A = 1 except possibly when the characteristic char(A) ≠ 2 and G has a cyclic Sylow 2-subgroup ≠ 1. In the exceptional case there is a unique (normal) subgroup H of G with index 2 and s B|A = s C|A where C = B H is the fixed ring. The remaining quadratic case is known and easily treated.

Parabolic Subgroups of SO2l Over a Dedekind Ring of Arithmetic Type

Let R be a commutative ring all of whose proper factor rings are finite and such that there exists a unit of infinite order. We show that for a subgroup P in G = SO(2l, R), l ≥ 3, containing the Borel subgroup B, the following alternative holds: either P contains a relative elementary subgroup E I for some ideal I = 0 or H is contained in a proper standard parabolic subgroup. For Dedekind rings of arithmetic type this allows us, under some mild additional assumptions on units, to describe completely the overgroups of B in G. Earlier, similar results for the special linear and symplectiv groups were obtained by A. V. Alexandrov and the second author. The proofs in the present paper follow the same general strategy, but are noticeably harder, from a technical viewpoint. Bibliography: 34 titles.

Homological stabilization for dedekind rings of the arithmetic type

We study the problem of stabilization in the higher K -theory of rings and improved stabilization. The triviality of the group of standard cycles is established in the case of rings of the arithmetic type. Some applications of the obtained results to the problem of homological stabilization are presented.

Grobner bases over a dual valuation domain

Buchberger’s algorithm was already studied over many kinds of rings such as principal ideal rings, noetherian valuation rings with zero divisors, Dedekind rings with zero divisors, Gaussian rings, ... . In this paper, we propose the Buchberger’s algorithm over V[e] satisfying to e 2 = 0 where V is any noetherian valuation domain and we give some applications in Zp [e] where p is a prime number and Zp = { a | a ∈ Z ,b / ∈ pZ}.

Pushout of quasi-finite and flat group schemes over a Dedekind ring

Let G, G1 and G2 be quasi-finite and flat group schemes over a complete discrete valuation ring R, φ1:G→G1 any morphism of R-group schemes and φ2:G→G2 a model map. We construct the pushout P of G1 and G2 over G in the category of R-affine group schemes. In particular when φ1 is a model map too we show that P is still a model of the generic fibre of G. We also provide a short proof for the existence of cokernels and quotients of finite and flat group schemes over any Dedekind ring.

Generating a power basis over a Dedekind ring

Let R be a Dedekind ring, K its quotient field, L a separable finite extension over K , and O L the integral closure of R in L . In this paper we provide a “practical” criterion that tests when a given α ∈ O L generates a power basis for O L over R (i.e. when O L = R [ α ] ), improving significantly a result in this direction by M. Charkani and O. Lahlou. Applications in the context of cyclotomic, quadratic, biquadratic number fields, and some Dedekind rings are provided.

On two sequences of sets of mappings of abstract sets into a Dedekind ring

This paper establishes some interrelations between a finite sequence of sets of mappings of an abstract set S into complete residue systems of pairwise relatively prime elements of any Dedekind ring and the corresponding sequence of sets of mappings of the set S into the complete residue system corresponding to the product of these elements. A relationship is revealed between the established interrelation and Lang's theorem on isomorphic factor rings. A string model is presented that is an interpretation of structures investigated in the cases of the ring of integers and a one-element set S. It is shown that the results obtained can be used for computing the number of combinatorial objects defined in terms of finite residue rings.

Hereditarily pure associative algebras over a Dedekind ring whose maximal ideals have finite indices

The concepts of purity and of weak purity play an important part in Abelian group theory. It turns out that these concepts may be approached by using the theory of varieties of groups. The idea behind the approach allowed their analogs to be defined for arbitrary algebras and made it possible to formulate major problems of research in this direction (see [1]). Within these frames, the analogs mentioned were explored for modules [2-4], universal algebras [5-7], semigroups [8-12], Lie algebras over an arbitrary associative commutative ring with unity [13], and associative rings [14]. In so doing, more attention was centered on the concept of atomic purity, an analog of weak purity. In fact, atomic purity seems to be more natural to be dealt with in studying a good many algebras, at least at the initial stage. In the present paper, we will look at this concept as applied to associative algebras over a Dedekind ring. (In our argument, atomic purity will be referred to merely as purity.) Namely, we will work to tackle the problem of describing hereditarily pure algebras [1, Problem 17] for a variety of all algebras over a Dedekind ring R whose maximal ideals have finite indices. (In what follows, unless otherwise stated, we retain the designation R for a ring with the property specified.) Note that our reasoning covers the case of algebras over a finite field and the case of associative rings, and incidentally, refines a result announced in [14]. We give necessary definitions and designations. The letter p always stands for a prime, while k, m, and n are used to denote nonnegative integers. The cardinal number of a set M is denoted by |M |. Throughout the paper, a module is a left unitary R-module; an algebra is an associative

Some Characterizations of Dedekind Rings

In this article we establish some characterizations for Dedekind rings.

Index-modules and applications

Let K be a commutative field, \({A\subseteq K}\) be a Dedekind ring and V be a K-vector space. For any pair of A-lattices R ≠ 0 and S of V, we define an A-submodule \({\left[R : S\right]^{\prime}_{A}}\) of K, their A-index-module. Once the basic properties of these modules are stated, we show that this notion can be used to recover more usual ones: the group-index, the relative invariant, the Fitting ideal of R/S when \({S\subseteq R}\), and the generalized index of Sinnott. As an example, we consider the following situation. Let F/k be a finite abelian extension of global function fields, with Galois group G, and degree g. Let ∞ be a place of k which splits completely in F/k. Let \({{\mathcal O}_{F}}\) be the ring of functions of F, which are regular outside the places of F sitting over ∞. Then one may use Stark units to define a subgroup \({\mathcal E_F}\) of \({{\mathcal O}_{F}^{\times}}\), the group of units of \({\mathcal O_F}\). We use the notion of index-module to prove that for every nontrivial irreducible rational character ψ of G, the ψ-part of \({\mathbb Z\left[g^{-1}\right]\otimes_\mathbb Z\left(\mathcal O_F^\times/\mathcal E_F\right)}\) and the ψ-part of \({\mathbb Z\left[g^{-1}\right]\otimes_\mathbb ZCl(\mathcal O_F)}\) have the same order.

Parabolic subgroups of SL n and Sp2l over a Dedekind ring of arithmetic type

Let R be a commutative ring all of whose proper factor rings are finite and in which a unit of infinite order exists. It is shown that for a subgroup P in G =SL(n, R), n ≥3,or in G =Sp(2l, R), l ≥2, containing the standard Borel subgroup B, the following alternative holds: either P contains a relative elementary subgroup EI for some ideal I≠0 or H is contained in a proper standard parabolic subgroup. For Dedekind rings of arithmetic type, this allows one, under some mild additional assumptions on units, to completely describe the overgroups of B in G. Bibliography: 30 titles.

Dynamical Gröbner bases over Dedekind rings

In this paper, we extend the notion of “dynamical Gröbner bases” introduced by the second author to Dedekind rings (with zero divisors). As an application, we dynamically solve the ideal membership problem and compute a generating set for the syzygy module over multivariate polynomial rings with coefficients in Dedekind rings. We also give a partial positive answer to a conjecture about Gröbner rings.

Subfields of the adele ring

We establish a natural one-to-one correspondence between immediate extensions of a Dedekind ring and subfields of its ring of adeles containing its ring of fractions. As a consequence of the existence theorems for immediate extensions proved previously, we state that e-closed fields within the class of countable subfields of the classical adele ring are surprising extensions of the field of rationals.

Schmidt modules and some of their applications

Let R be an associative ring with unit. A nonsemisimple right R-module M = M R is referred to as a (right) Schmidt module if every proper (right) submodule in M is semisimple, and a module M is called a (right) generalized Schmidt module if M is not a Schmidt module and each of its proper (right) submodule is either a semisimple module or a Schmidt module. A left Schmidt R-module and a left generalized Schmidt R-module are defined similarly. In the paper, a complete description of the structure of right Schmidt R-modules and generalized Schmidt R-modules is given, the existence of Schmidt R-submodules in any nonsemisimple Artinian module is established, and a complete description of nonsemisimple Artinian modules in which every Schmidt submodule is distinguished as a direct summand is presented. As corollaries, characterizations of (generalized) Schmidt modules over a Dedekind ring and over a matrix ring over this ring are obtained in the paper.