Units of group rings, semigroup rings and orders in finite dimensional algebras

• Grant number: 13/14824-5
• Support Opportunities: Scholarships abroad - Research
• Start date: February 03, 2014
• End date: February 02, 2015
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Edson Ryoji Okamoto Iwaki
• Grantee: Edson Ryoji Okamoto Iwaki
• Host Investigator: Eric Jespers
• Host Institution: Centro de Matemática, Computação e Cognição (CMCC). Universidade Federal do ABC (UFABC). Ministério da Educação (Brasil). Santo André , SP, Brazil
• Institution abroad: Vrije Universiteit Brussel (VUB), Belgium

Abstract

The hyperbolic groups were defined by Gromov in [17], from the concept of hyperbolic metric space. Given a group finitely generated G, it is possible to construct a metric that associated with the Cayley graph of G defines a metric space. The group G is hyperbolic if its Cayley graph is a hyperbolic metric space. Let G be a group, let R be an associative ring with unity. RG denote the set of all formal finite sums of elements r_gg, g G, r_g R, i.e,\sum_ {g G} r_gg. The subsetU (RG) = {u RG | uv = vu = 1 for some v RG}is called the group of units of RG. The unit group U(ZG) is an object of intense research in Group Rings. For a finite group G, we are interested in studying: * The hyperbolicity of groups U (RG) and certain finite-dimensional algebras. * Investigate the structure of the unit group U(RG). An important result that justifies the study of the first two questions above is the fact that the unit group U (ZG) is finitely generated, when G is finite. Following the methods of [61], this study is possible for rings R \ not = Z and other structures for the object G as semigroups. (AU)