Ultrafunctor

1975 ◽  
Vol 27 (2) ◽  
pp. 372-375 ◽  
Author(s):  
Andy R. Magid
Keyword(s):  
Finite Fields ◽  
Abelian Groups ◽  
Commutative Rings ◽  
Picard Group ◽  
Brauer Group

Let G be a functor from commutative rings to abelian groups and let ﹛Rt : i ∈ S﹜ be a family of commutative rings indexed by the set S. Let be an ultrafilter on S, and let denote the ultraproduct of the Rt with respect to . This paper studies the problem of computing from the G(Rj) via the mapThe functors studied are Pic = Picard group, Br = Brauer group, U = units, and the functors K0, K1, SK1, K2 of Algebraic K-Theory. For G = Pic, U, K1 and SK1, (*) is always a monomorphism. An example is given to show that even if all the Rt are finite fields the map (*) has a kernel for G = K2.

scholarly journals Embedding the Picard group inside the class group: the case of $$\mathbb {Q}$$-factorial complete toric varieties

2021 ◽  
Author(s):  
Michele Rossi ◽  
Lea Terracini
Keyword(s):  
Free Group ◽  
Toric Variety ◽  
Class Group ◽  
Abelian Groups ◽  
Toric Varieties ◽  
Picard Group ◽  
Closed Field ◽  
Finitely Generated ◽  
Free Part ◽  
Canonical Injection

AbstractLet X be a $$\mathbb {Q}$$ Q -factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group $$\mathrm{Pic}(X)$$ Pic ( X ) in the group $$\mathrm{Cl}(X)$$ Cl ( X ) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $$\mathrm{Pic}(X)$$ Pic ( X ) in $$\mathrm{Cl}(X)$$ Cl ( X ) is contained in a free part of the latter group.

scholarly journals Uniform distribution of sequences in rings of integral matrices

1979 ◽  
Vol 20 (2) ◽  
pp. 169-178
Author(s):  
Harald Niederreiter ◽  
Jau-Shyong Shiue
Keyword(s):  
Uniform Distribution ◽  
Finite Fields ◽  
Commutative Rings ◽  
Algebraic Integers ◽  
Matrix Rings ◽  
Integral Matrices ◽  
Rings Of Algebraic Integers ◽  
Uniform Distribution Of Sequences ◽  
Satisfactory Theory ◽  
Noncommutative Setting

For various discrete commutative rings a concept of uniform distribution has already been introduced and studied, for example, for the ring of rational integers by Niven [9] (see also Kuipers and Niederreiter [2, Ch. 5]), for the rings of Gaussian and Eisenstein integers by Kuipers, Niederreiter, and Shiue [3], for rings of algebraic integers by Lo and Niederreiter [4], [7], and for finite fields by Gotusso [1] and Niederreiter and Shiue [8]. In the present paper, we shall show that a satisfactory theory of uniform distribution can also be developed in a noncommutative setting, namely for matrix rings over the rational integers.

On 2-Groups as Galois Groups

1995 ◽  
Vol 47 (6) ◽  
pp. 1253-1273 ◽  
Author(s):  
Arne Ledet
Keyword(s):  
Galois Group ◽  
Direct Product ◽  
Galois Extension ◽  
Quaternion Algebra ◽  
Abelian Groups ◽  
Galois Groups ◽  
Embedding Problem ◽  
Brauer Group ◽  
Characteristic 2 ◽  
Cyclic Kernel

AbstractLet L/K be a finite Galois extension in characteristic ≠ 2, and consider a non-split Galois theoretical embedding problem over L/K with cyclic kernel of order 2. In this paper, we prove that if the Galois group of L/K is the direct product of two subgroups, the obstruction to solving the embedding problem can be expressed as the product of the obstructions to related embedding problems over the corresponding subextensions of L/K and certain quaternion algebra factors in the Brauer group of K. In connection with this, the obstructions to realising non-abelian groups of order 8 and 16 as Galois groups over fields of characteristic ≠ 2 are calculated, and these obstructions are used to consider automatic realisations between groups of order 4, 8 and 16.

scholarly journals A Study on Finite Abelian Groups: Sylow’s Theorems Based

2017 ◽  
Vol 1 (3) ◽  
Author(s):  
Amaira Moaitiq Mohammed Al-Johani
Keyword(s):  
Abelian Group ◽  
Algebraic Structure ◽  
Abelian Groups ◽  
Commutative Rings ◽  
Abstract Algebra ◽  
Vector Spaces ◽  
Finite Abelian Groups ◽  
Study Objective ◽  
Group Operation ◽  
Basic Concepts

In abstract algebra, an algebraic structure is a set with one or more finitary operations defined on it that satisfies a list of axioms. Algebraic structures include groups, rings, fields, and lattices, etc. A group is an algebraic structure (????, ∗), which satisfies associative, identity and inverse laws. An Abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, these are the groups that obey the axiom of commutatively. The concept of an Abelian group is one of the first concepts encountered in abstract algebra, from which many other basic concepts, such as rings, commutative rings, modules and vector spaces are developed. This study sheds the light on the structure of the finite abelian groups, basis theorem, Sylow’s theorem and factoring finite abelian groups. In addition, it discusses some properties related to these groups. The researcher followed the exploratory and comparative approaches to achieve the study objective. The study has shown that the theory of Abelian groups is generally simpler than that of their non-abelian counter parts, and finite Abelian groups are very well understood.  

scholarly journals Automorphism groups and Picard groups of additive full subcategories

2010 ◽  
Vol 107 (1) ◽  
pp. 5 ◽  
Author(s):  
Naoya Hiramatsu ◽  
Yuji Yoshino
Keyword(s):  
Automorphism Group ◽  
Direct Product ◽  
Full Subcategory ◽  
Automorphism Groups ◽  
Commutative Rings ◽  
Picard Group ◽  
Picard Groups

We study category equivalences between additive full subcategories of module categories over commutative rings. And we are able to define the Picard group of additive full subcategories. The aim of this paper is to study the properties of the Picard groups and show that the automorphism group of an additive full subcategory is a semi-direct product of the Picard group with the group of algebra automorphisms of the ring.

scholarly journals On Group Codes Over Elementary Abelian Groups

2003 ◽  
Vol 8 (2) ◽  
pp. 145
Author(s):  
Adnan A. Zain
Keyword(s):  
Abelian Group ◽  
Finite Fields ◽  
Linear Codes ◽  
Abelian Groups ◽  
Parity Check ◽  
Group Codes ◽  
New Codes ◽  
Generator Matrices

For group codes over elementary Abelian groups we present definitions of the generator and the parity check matrices, which are matrices over the ring of endomorphism of the group. We also lift the theorem that relates the parity check and the generator matrices of linear codes over finite fields to group codes over elementary Abelian groups. Some new codes that are MDS, self-dual, and cyclic over the Abelian group with four elements are given.

scholarly journals FINITE QUASI-FROBENIUS MODULES AND LINEAR CODES

2004 ◽  
Vol 03 (03) ◽  
pp. 247-272 ◽  
Author(s):  
MARCUS GREFERATH ◽  
ALEXANDR NECHAEV ◽  
ROBERT WISBAUER
Keyword(s):  
Finite Fields ◽  
Linear Codes ◽  
General Setting ◽  
Commutative Rings ◽  
Finite Ring ◽  
Weight Enumerators ◽  
Frobenius Rings ◽  
Finite Frobenius Rings

The theory of linear codes over finite fields has been extended by A. Nechaev to codes over quasi-Frobenius modules over commutative rings, and by J. Wood to codes over (not necessarily commutative) finite Frobenius rings. In the present paper, we subsume these results by studying linear codes over quasi-Frobenius and Frobenius modules over any finite ring. Using the character module of the ring as alphabet, we show that fundamental results like MacWilliams' theorems on weight enumerators and code isometry can be obtained in this general setting.

scholarly journals Commutative rings with homomorphic power functions

1992 ◽  
Vol 15 (1) ◽  
pp. 91-102
Author(s):  
David E. Dobbs ◽  
John O. Kiltinen ◽  
Bobby J. Orndorff
Keyword(s):  
Commutative Ring ◽  
Finite Fields ◽  
Structure Theorem ◽  
Commutative Rings ◽  
Prime Characteristic ◽  
Power Functions ◽  
Finite Case

A (commutative) ringR(with identity) is calledm-linear (for an integerm≥2) if(a+b)m=am+bmfor allaandbinR. Them-linear reduced rings are characterized, with special attention to the finite case. A structure theorem reduces the study ofm-linearity to the case of prime characteristic, for which the following result establishes an analogy with finite fields. For each primepand integerm≥2which is not a power ofp, there exists an integers≥msuch that, for each ringRof characteristicp,Rism-linear if and only ifrm=rpsfor eachrinR. Additional results and examples are given.

scholarly journals Elementary equivalence for finitely generated nilpotent groups and multilinear maps

1998 ◽  
Vol 58 (3) ◽  
pp. 479-493 ◽  
Author(s):  
Francis Oger
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Elementary Equivalence ◽  
Linear Map ◽  
Multilinear Maps

We show that two finitely generated finite-by-nilpotent groups are elementarily equivalent if and only if they satisfy the same sentences with two alternations of quantifiers. For each integer n ≥ 2, we prove the same result for the following classes of structures:(1) the (n + 2)-tuples (A1, …, An+1, f), where A1, …, An+1 are disjoint finitely generated Abelian groups and f: A1 × … × An → An+1 is a n-linear map;(2) the triples (A, B, f), where A, B are disjoint finitely generated Abelian groups and f: An → B is a n-linear map;(3) the pairs (A, f), where A is a finitely generated Abelian group and f: An → A is a n-linear map.In the proof, we use some properties of commutative rings associated to multilinear maps.

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