The problem with which this investigation is concerned is that of determining the properties of three radicals defined on an arbitrary ring and determining when these radicals coincide. The three radicals discussed are the nil radical, the Jacobsson radical, and the Brown-McCoy radical.

The main purpose of this paper is to examine the road towards the structure of simple and semi-simple Artinian rings. We refer to these structure theorems as the Wedderburn-Artin theorems. On this journey, we will discuss R-modules, the Jacobson radical, Artinian rings, nilpotency, idempotency, and more. Once we reach our destination, we will examine some implications of these theorems. As a fair warning, no ring will be assumed to be commutative, or to have unity. On that note, the reader should be familiar with the basic findings from Group Theory and Ring Theory.

The Jacobson radical of a ring was first formally studied in 1945 by Nathan Jacobson and is an important object in modern abstract algebra. The analogous notion of the Jacobson radical for a module is referred to as the radical of a module. The radical of a module is the intersection of all its maximal submodules. In general, the radical of a module is simpler than the module itself and contains important information about the module. The study of the radical of a module often appears as an incidental to other investigations. This thesis represents work towards understanding the radical of a module extension. Given a ring $R$ and $R$-modules $A,B,X$ such that $X$ is an extension of $B$ by $A$ as in the short exact sequence $$0 rightarrow A rightarrow X rightarrow B rightarrow 0 ,$$ we seek to determine properties of the radical of $X$, denoted $rad{X}$. These properties are dependent on the ring $R$ and properties of the modules $A$ and $B$. In this thesis we examine several different types of extensions and discuss a phenomenon in which a non-zero radical implies a split sequence. We work in the context of rings and their ideals. Extensions of abelian groups provide motivation for the results we prove about injectivity of radicals of extensions involving divisible modules and torsion modules. We are able to prove such properties of the radical for extensions of modules over principal ideal domains and Dedekind domains. Expanding upon these cases, we explore a more general construction of an extension and use it to explain our motivating abelian group results. We use the theorems proven about this construction to remark on possible generalizations to other types of rings and modules. We conclude with plans to generalize our statements by translating into terms of infinite matrices and $h$-local rings.

by Au Yun-Nam. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 133-137). / Introduction --- p.iv / Chapter 1 --- Preliminaries --- p.1 / Chapter 1.1 --- Some Semigroup Properties --- p.1 / Chapter 1.2 --- General Properties of Semigroup Algebras --- p.5 / Chapter 1.3 --- Group Algebras --- p.7 / Chapter 1.3.1 --- Some Basic Properties of Groups --- p.7 / Chapter 1.3.2 --- General Properties of Group Algebras --- p.8 / Chapter 1.3.3 --- Δ-Method for Group Algebras --- p.10 / Chapter 1.4 --- Graded Algebras --- p.12 / Chapter 1.5 --- Crossed Products and Smash Products --- p.14 / Chapter 2 --- Radicals of Graded Rings --- p.17 / Chapter 2.1 --- Jacobson Radical of Crossed Products --- p.17 / Chapter 2.2 --- Graded Radicals and Reflected Radicals --- p.18 / Chapter 2.3 --- Radicals of Group-graded Rings --- p.24 / Chapter 2.4 --- Algebras Graded by Semilattices --- p.26 / Chapter 2.5 --- Algebras Graded by Bands --- p.27 / Chapter 2.5.1 --- Hereditary Radicals of Band-graded Rings --- p.27 / Chapter 2.5.2 --- Special Band-graded Rings --- p.30 / Chapter 3 --- Radicals of Semigroup Algebras --- p.34 / Chapter 3.1 --- Radicals of Polynomial Rings --- p.34 / Chapter 3.2 --- Radicals of Commutative Semigroup Algebras --- p.36 / Chapter 3.2.1 --- Commutative Cancellative Semigroups --- p.37 / Chapter 3.2.2 --- General Commutative Semigroups --- p.39 / Chapter 3.2.3 --- The Nilness and Semiprimitivity of Commutative Semigroup Algebras --- p.45 / Chapter 3.3 --- Radicals of Cancellative Semigroup Algebras --- p.48 / Chapter 3.3.1 --- Group of Fractions of Cancellative Semigroups --- p.48 / Chapter 3.3.2 --- Jacobson Radical of Cancellative Semigroup Algebras --- p.54 / Chapter 3.3.3 --- Subsemigroups of Polycyclic-by-Finite Groups --- p.57 / Chapter 3.3.4 --- Nilpotent Semigroups --- p.59 / Chapter 3.4 --- Radicals of Algebras of Matrix type --- p.62 / Chapter 3.4.1 --- Properties of Rees Algebras --- p.62 / Chapter 3.4.2 --- Algebras Graded by Elementary Rees Matrix Semigroups --- p.65 / Chapter 3.5 --- Radicals of Inverse Semigroup Algebras --- p.68 / Chapter 3.5.1 --- Properties of Inverse Semigroup Algebras --- p.69 / Chapter 3.5.2 --- Radical of Algebras of Clifford Semigroups --- p.72 / Chapter 3.5.3 --- Semiprimitivity Problems of Inverse Semigroup Algebras --- p.73 / Chapter 3.6 --- Other Semigroup Algebras --- p.76 / Chapter 3.6.1 --- Completely Regular Semigroup Algebras --- p.76 / Chapter 3.6.2 --- Separative Semigroup Algebras --- p.77 / Chapter 3.7 --- Radicals of Pi-semigroup Algebras --- p.80 / Chapter 3.7.1 --- PI-Algebras --- p.80 / Chapter 3.7.2 --- Permutational Property and Algebras of Permutative Semigroups --- p.80 / Chapter 3.7.3 --- Radicals of PI-algebras --- p.82 / Chapter 4 --- Finiteness Conditions on Semigroup Algebras --- p.85 / Chapter 4.1 --- Introduction --- p.85 / Chapter 4.1.1 --- Preliminaries --- p.85 / Chapter 4.1.2 --- Semilattice Graded Rings --- p.86 / Chapter 4.1.3 --- Group Graded Rings --- p.88 / Chapter 4.1.4 --- Groupoid Graded Rings --- p.89 / Chapter 4.1.5 --- Semigroup Graded PI-Algebras --- p.91 / Chapter 4.1.6 --- Application to Semigroup Algebras --- p.92 / Chapter 4.2 --- Semiprime and Goldie Rings --- p.92 / Chapter 4.3 --- Noetherian Semigroup Algebras --- p.99 / Chapter 4.4 --- Descending Chain Conditions --- p.107 / Chapter 4.4.1 --- Artinian Semigroup Graded Rings --- p.107 / Chapter 4.4.2 --- Semilocal Semigroup Algebras --- p.109 / Chapter 5 --- Dimensions and Second Layer Condition on Semigroup Algebras --- p.119 / Chapter 5.1 --- Dimensions --- p.119 / Chapter 5.1.1 --- Gelfand-Kirillov Dimension --- p.119 / Chapter 5.1.2 --- Classical Krull and Krull Dimensions --- p.121 / Chapter 5.2 --- The Growth and the Rank of Semigroups --- p.123 / Chapter 5.3 --- Dimensions on Semigroup Algebras --- p.124 / Chapter 5.4 --- Second Layer Condition --- p.128 / Notations and Abbreviations --- p.132 / Bibliography --- p.133

O objetivo desta dissertação é estudar o radical de Jacobson de anéis de polinômios diferenciais. Mostramos um resultado de M. Ferrero, K. Kishimoro, K. Motose, que mostra que no caso geral, o radical de um anel de polinômios diferenciais é um anel de polinômios diferenciais sobre algum ideal do anel dos coeficientes. Assumindo que o anel dos coeficientes satisfaça uma identidade polinomial, mostramos seguindo B. Madill que este ideal é um ideal nil. Se o anel dos coeficientes é adicionalmente localmente nilpotente, seguindo J. Bell, B. Madill, F. Shinko, mostramos que o anel de polinômios diferenciais será localmente nilpotente. Ainda seguindo J. Bell et al, se o anel dos coeficientes é uma álgebra sobre um corpo de característica zero e tal álgebra satisfaz uma identidade polinomial, mostramos que o ideal nil é o radical de Köthe. Para tais demonstrações, cobriremos os tópicos preliminares necessários para entender os enunciados: radical nil, radical de Levitzki, radical de Baer, radical de Jacobson e propriedades, anéis PI, polinômios centrais, teorema de Kaplansky. / The aim of this work is to study the Jacobson radical of differential polynomial rings. We show a result of M. Ferrero, K. Kishimoto, K. Motose, which shows that in general, the radical of a differential polynomial ring is a differential polynomial ring over some ideal of the ring of coefficients. Assuming that the ring of coefficients satisfies a polynomial identity, we show following B. Madill that this ideal is nil. If the ring of coefficients is additionally locally nilpotent, following J. Bell, B. Madill, F. Shinko, we show that the differential polynomial ring is locally nilpotent. Still following J. Bell et al, if the ring of coefficients is an algebra over a field of zero characteristic and this algebra satisfies a polynomial identity, we show that the nil ideal is the Köthe radical. For the proofs, we cover the preliminary topics necessary for understanding the statements: nil radical, Levitzki radical, Baer radical, Jacobson radical and its properties, PI-rings, central polynomials, Kaplanskys theorem.

O objetivo desta dissertação é estudar o radical de Jacobson de anéis de polinômios diferenciais. Mostramos um resultado de M. Ferrero, K. Kishimoro, K. Motose, que mostra que no caso geral, o radical de um anel de polinômios diferenciais é um anel de polinômios diferenciais sobre algum ideal do anel dos coeficientes. Assumindo que o anel dos coeficientes satisfaça uma identidade polinomial, mostramos seguindo B. Madill que este ideal é um ideal nil. Se o anel dos coeficientes é adicionalmente localmente nilpotente, seguindo J. Bell, B. Madill, F. Shinko, mostramos que o anel de polinômios diferenciais será localmente nilpotente. Ainda seguindo J. Bell et al, se o anel dos coeficientes é uma álgebra sobre um corpo de característica zero e tal álgebra satisfaz uma identidade polinomial, mostramos que o ideal nil é o radical de Köthe. Para tais demonstrações, cobriremos os tópicos preliminares necessários para entender os enunciados: radical nil, radical de Levitzki, radical de Baer, radical de Jacobson e propriedades, anéis PI, polinômios centrais, teorema de Kaplansky. / The aim of this work is to study the Jacobson radical of differential polynomial rings. We show a result of M. Ferrero, K. Kishimoto, K. Motose, which shows that in general, the radical of a differential polynomial ring is a differential polynomial ring over some ideal of the ring of coefficients. Assuming that the ring of coefficients satisfies a polynomial identity, we show following B. Madill that this ideal is nil. If the ring of coefficients is additionally locally nilpotent, following J. Bell, B. Madill, F. Shinko, we show that the differential polynomial ring is locally nilpotent. Still following J. Bell et al, if the ring of coefficients is an algebra over a field of zero characteristic and this algebra satisfies a polynomial identity, we show that the nil ideal is the Köthe radical. For the proofs, we cover the preliminary topics necessary for understanding the statements: nil radical, Levitzki radical, Baer radical, Jacobson radical and its properties, PI-rings, central polynomials, Kaplanskys theorem.

Cette thèse s'attache à décrire des algorithmes qui répondent à des questions provenant de la théorie des anneaux et des modules. Nous restreindrons essentiellement notre étude à des algorithmes déterministes, en temps polynomial, ainsi qu'aux anneaux et modules finis. Le premier des principaux résultats de cette thèse concerne le problème de l'isomorphisme entre modules : nous décrivons deux algorithmes distincts qui, étant donnée un anneau fini R et deux R-modules M et N finis, déterminent si M et N sont isomorphes. S'ils le sont, les deux algorithmes exhibent un tel isomorphisme. De plus, nous montrons comment calculer un ensemble de générateurs de taille minimale pour un module donné, et comment construire des couvertures projectives et des enveloppes injectives. Nous décrivons ensuite des tests mettant en évidence le caractère simple, projectif ou injectif d'un module, ainsi qu'un test constructif de l'existence d'un homomorphisme demodules surjectif entre deux modules finis, l'un d'entre eux étant projectif. Par contraste, nous montrons le résultat négatif suivant : le problème consistant à tester l'existence d'un homomorphisme de modules injectif entre deux modules, l'un des deux étant projectif, est NP-complet.La dernière partie de cette thèse concerne le problème de l'approximation du radical de Jacobson d'un anneau fini. Il s'agit de déterminer un idéal bilatère nilpotent tel que l'anneau quotient correspondant soit \presque" semi-simple. La notion de \semi-simplicité approchée" que nous utilisons est la séparabilité. / In this thesis we are interested in describing algorithms that answer questions arising in ring and module theory. Our focus is on deterministic polynomial-time algorithms and rings and modules that are finite. The first main result of this thesis concerns the module isomorphism problem: we describe two distinct algorithms that, given a finite ring R and two finite R-modules M and N, determine whether M and N are isomorphic. If they are, the algorithms exhibit such a isomorphism. In addition, we show how to compute a set of generators of minimal cardinality for a given module, and how to construct projective covers and injective hulls. We also describe tests for module simplicity, projectivity, and injectivity, and constructive tests for existence of surjective module homomorphisms between two finite modules, one of which is projective. As a negative result, we show that the problem of testing for existence of injective module homomorphisms between two finite modules, one of which is projective, is NP-complete. The last part of the thesis is concerned with finding a good working approximation of the Jacobson radical of a finite ring, that is, a two-sided nilpotent ideal such that the corresponding quotient ring is \almost" semisimple. The notion we use to approximate semisimplicity is that of separability.