The research field of graded ring theory is a rich area of mathematics with many connections to e.g. the field of operator algebras. In the last 15 years, algebraists and operator algebraists have defined algebraic analogues of important operator algebras. Some of those analogues are rings that come equipped with a group grading. We want to reach a better understanding of the graded structure of those analogue rings. Among group graded rings, the strongly graded rings stand out as being especially well-behaved. The development of the general theory of strongly graded rings was initiated by Dade in the 1980s and since then numerous structural results have been established for strongly graded rings.  In this thesis, we study the class of epsilon-strongly graded rings which was recently introduced by Nystedt, Öinert and Pinedo. This class is a natural generalization of the well-studied class of unital strongly graded rings. Our aim is to lay the foundation for a general theory of epsilon-strongly graded rings generalizing the theory of strongly graded rings. This thesis is based on three articles. The first two articles mainly concern structural properties of epsilon-strongly graded rings. In the first article, we investigate a functorial construction called the induced quotient group grading. In the second article, using results from the first article, we generalize the Hilbert Basis Theorem for strongly graded rings to epsilon-strongly graded rings and apply it to Leavitt path algebras.  In the third article, we study the graded structure of algebraic Cuntz-Pimsner rings. In particular, we obtain a partial classification of unital strongly, epsilon-strongly and nearly epsilon-strongly graded Cuntz-Pimsner rings up to graded isomorphism.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In representation theory, having a representation of a group G is equivalent to having a kG-module. Since |G-modules which are sums of irreducible kG-modules form a very important class in the theory of modules, to know conditions for a kG-module be irreducible or completely reducible from the particularities of the field k and the group G become a very important issue, whose solution was originally presented by the German mathematician Heinrich Maschke which proved that if the order of G is not a multiple of the characteristic of the field k, then kG is completely reducible (or semisimple). From there, issues unrelated to representation theory, but that concern the semisimplicity of cross products in general are treated as Maschke-type theorem. Our goal in this dissertation is to present some versions of this theorem, starting with classic versions involving cross products for actions of groups on algebras and then versions for Hopf algebras and smash products. / Na teoria de representações de grupos, ter uma representação de um grupo G é equivalente a ter um kG-módulo. Desde que kG-módulos que são somas de kG-módulos irredutíveis formam uma classe bastante importante na teoria de módulos, conhecer condições para que um kG-módulo seja irredutível ou completamente redutível a partir das particularidades do corpo k e do grupo G passou a ser um problema bastante importante. Problema este cuja solução foi originalmente apresentada pelo matemático alemão Heinrich Maschke que provou que se a ordem do grupo G não for múltiplo da característica do corpo k, então kG é completamente redutível (ou semissimples). A partir daí, questões independentes a teoria de representações, mas que dizem respeito a semissimplicidade de produtos cruzados em geral são tratados como Teorema tipo-Maschke. Nosso objetivo neste trabalho é apresentar algumas versões deste teorema. Iniciamos com versões mais clássicas envolvendo produtos cruzados globais e parciais para em seguida estudarmos versões em álgebras de Hopf e produtos smash.

It is well known that communication systems employing multiple transmit and multiple receive antennas provide high data rates along with increased reliability. It has been shown that coding across both spatial and temporal domains together, called Space-Time Coding (STC), achieves, a diversity order equal to the product of the number of transmit and receive antennas. Space-Time Block Codes (STBC) achieving the maximum diversity is called full-diversity STBCs. An STBC is called information-lossless, if the structure of it is such that the maximum mutual information of the resulting equivalent channel is equal to the capacity of the channel. This thesis deals with high-rate and information-lossless STBCs obtained from certain matrix algebras called Crossed-Product Algebras. First we give constructions of high-rate STBCs using both commutative and non-commutative matrix algebras obtained from appropriate representations of extensions of the field of rational numbers. In the case of commutative algebras, we restrict ourselves to fields and call the STBCs obtained from them as STBCs from field extensions. In the case of non-commutative algebras, we consider only the class of crossed-product algebras. For the case of field extensions, we first construct high-rate; full-diversity STBCs for arbitrary number of transmit antennas, over arbitrary apriori specified signal sets. Then we obtain a closed form expression for the coding gain of these STBCs and give a tight lower bound on the coding gain of some of these STBCs. This lower bound in certain cases indicates that some of the STBCs from field extensions are optimal m the sense of coding gain. We then show that the STBCs from field extensions are information-lossy. However, we also show that the finite-signal-set capacity of the STBCs from field extensions can be improved by increasing the symbol rate of the STBCs. The simulation results presented show that our high-rate STBCs perform better than the rate-1 STBCs in terms of the bit error rate performance. Then we proceed to present a construction of high-rate STBCs from crossed-product algebras. After giving a sufficient condition on the crossed-product algebras under which the resulting STBCs are information-lossless, we identify few classes of crossed-product algebras that satisfy this sufficient condition and also some classes of crossed-product algebras which are division algebras which lead to full-diversity STBCs. We present simulation results to show that the STBCs from crossed-product algebras perform better than the well-known codes m terms of the bit error rate. Finally, we introduce the notion of asymptotic-information-lossless (AILL) designs and give a necessary and sufficient condition under which a linear design is an AILL design. Analogous to the condition that a design has to be a full-rank design to achieve the point corresponding to the maximum diversity of the optimal diversity-multiplexing tradeoff, we show that a design has to be AILL to achieve the point corresponding to the maximum multiplexing gain of the optimal diversity-multiplexing tradeoff. Using the notion of AILL designs, we give a lower bound on the diversity-multiplexing tradeoff achieved by the STBCs from both field extensions and division algebras. The lower bound for STBCs obtained from division algebras indicates that they achieve the two extreme points, 1 e, zero multiplexing gain and zero diversity gain, of the optimal diversity-multiplexing tradeoff. Also, we show by simulation results that STBCs from division algebras achieves all the points on the optimal diversity-multiplexing tradeoff for n transmit and n receive antennas, where n = 2, 3, 4.