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Concept# Ring (mathematics)

Summary

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.
Formally, a ring is an abelian group whose operation is called addition, with a second binary operation called multiplication that is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors use the term "" with a missing "i" to refer to the more general structure that omits this last requirement; see .)
Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the result) has profound implications on

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Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus

Commutative ring

In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebr

Module (mathematics)

In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module generalizes also the notion of abelian group, sin

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MATH-310: Algebra

Study basic concepts of modern algebra: groups, rings, fields.

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Algebraic geometry is the common language for many branches of modern research in mathematics. This course gives an introduction to this field by studying algebraic curves and their intersection theory.

MATH-311: Rings and modules

The students are going to solidify their knowledge of ring and module theory with a major emphasis on commutative algebra and a minor emphasis on homological algebra.

The Grothendieck group of permutation modules for a finite group G becomes a biset functor when G is allowed to vary. All the composition factors of this biset functor are determined.

This thesis is concerned with computations of bounds for two different arithmetic invariants. In both cases it is done with the intention of proving some algebraic or arithmetic properties for number fields. The first part is devoted to computations of lower bounds for the Lenstra's constant. For a number field K the Lenstra's constant is denoted Λ(K) and defined as the length of the largest exceptional sequence in K. An exceptional sequence is a set of units in K such that for any two among them their difference is a unit as well. H.W. Lenstra showed that if Λ(K) is large enough – bigger than a constant depending on the degree and the discriminant of K – then the ring of integers of K is Euclidean with respect to the norm. Using computer software PARI/GP and some algorithms from graph theory we construct exceptional sequences in number fields having a small discriminant. These exceptional sequences yield lower bounds for Lenstra's constant which are large enough to prove the existence of 42 new Euclidean number fields of degree 8 to 12. The aim of the second part of this thesis is proving upper bounds for the torsion part of the K-groups of a number field ring of integers. A method due to C. Soulé yields bounds for the torsion of these K-groups depending on an invariant of hermitian lattices over number fields. Firstly we describe some properties of rank one hermitian lattices, especially of ideal lattices. Secondly we apply these properties to arbitrary rank hermitian lattices and this implies a significant improvement of the upper bounds for their invariants and accordingly for the torsion of K-groups. The progress mainly achieves much lower contributions of the number field attributes, particularly the degree and the absolute discriminant.

Let k be an algebraically closed field of characteristic p, where p is a prime number or 0. Let G be a finite group and ppk(G) be the Grothendieck group of p-permutation kG-modules. If we tensor it with C, then Cppk becomes a C-linear biset functor. Recall that the simple biset functor SH,V are parametrized by pairs (H,V), where H is a finite group and V a simple COut(H)-module. If we only consider p'-groups, then Cppk = CRk is the usual representation functor and we know the simple functors which are its composition factors. If we consider only p-groups, then Cppk = CB is the Burnside functor and we also know the simple functors which are its composition factors. We want to find the composition factors of Cppk in general. In order to achieve this, we first show that the composition factors from the special cases above are also composition factors for Cppk. Then, we consider groups of little order and try to find new composition factors. This leads us to find the following new composition factors : The simple factors SCm,Cξ and SCp×Cp× Cm,Cξ, where (m,ξ) runs over the set of all pairs formed by a positive integer m prime to p and a primitive character ξ : (Z/mZ)* → C*. Their multiplicity as composition factors is 1. The simple factors SCp⋊Cl, C, where l is a number prime to p, the action of Cl on Cp is faithful and C is the trivial COut(Cp ⋊ Cl)-module. Their multiplicity as composition factors is φ(l). The simple functors SG,C, where G is a finite p-hypo-elementary B-group (for which an explicit classification is done) and C the trivial COut(G)-module. We also show that some specific simple functors appear, indexed by the groups C3 ⋊ C4, C5 ⋊ C4 and A4. On the way, we find all the composition factors of the subfunctor of permutation modules.