Vizsgálatok algebrai struktúrák radikálelméletében (I) = Investigations in radical theory of algebraic structures. I.

The paper consists of the first two chapters and of the references of the author's D.Sc. thesis. The thesis was compiled from some 50 papers of the author, each of which has appeared and been reviewed in Mathematical Reviews. Four more chapters of the thesis will appear (\Investigations in radical theory of algebraic structures" (Hungarian)

The radicals of semigroup algebras with chain conditions.

by Au Yun-Nam.Thesis (M.Phil.)--Chinese University of Hong Kong, 1996.Includes bibliographical references (leaves 133-137).Introduction --- p.ivChapter 1 --- Preliminaries --- p.1Chapter 1.1 --- Some Semigroup Properties --- p.1Chapter 1.2 --- General Properties of Semigroup Algebras --- p.5Chapter 1.3 --- Group Algebras --- p.7Chapter 1.3.1 --- Some Basic Properties of Groups --- p.7Chapter 1.3.2 --- General Properties of Group Algebras --- p.8Chapter 1.3.3 --- Δ-Method for Group Algebras --- p.10Chapter 1.4 --- Graded Algebras --- p.12Chapter 1.5 --- Crossed Products and Smash Products --- p.14Chapter 2 --- Radicals of Graded Rings --- p.17Chapter 2.1 --- Jacobson Radical of Crossed Products --- p.17Chapter 2.2 --- Graded Radicals and Reflected Radicals --- p.18Chapter 2.3 --- Radicals of Group-graded Rings --- p.24Chapter 2.4 --- Algebras Graded by Semilattices --- p.26Chapter 2.5 --- Algebras Graded by Bands --- p.27Chapter 2.5.1 --- Hereditary Radicals of Band-graded Rings --- p.27Chapter 2.5.2 --- Special Band-graded Rings --- p.30Chapter 3 --- Radicals of Semigroup Algebras --- p.34Chapter 3.1 --- Radicals of Polynomial Rings --- p.34Chapter 3.2 --- Radicals of Commutative Semigroup Algebras --- p.36Chapter 3.2.1 --- Commutative Cancellative Semigroups --- p.37Chapter 3.2.2 --- General Commutative Semigroups --- p.39Chapter 3.2.3 --- The Nilness and Semiprimitivity of Commutative Semigroup Algebras --- p.45Chapter 3.3 --- Radicals of Cancellative Semigroup Algebras --- p.48Chapter 3.3.1 --- Group of Fractions of Cancellative Semigroups --- p.48Chapter 3.3.2 --- Jacobson Radical of Cancellative Semigroup Algebras --- p.54Chapter 3.3.3 --- Subsemigroups of Polycyclic-by-Finite Groups --- p.57Chapter 3.3.4 --- Nilpotent Semigroups --- p.59Chapter 3.4 --- Radicals of Algebras of Matrix type --- p.62Chapter 3.4.1 --- Properties of Rees Algebras --- p.62Chapter 3.4.2 --- Algebras Graded by Elementary Rees Matrix Semigroups --- p.65Chapter 3.5 --- Radicals of Inverse Semigroup Algebras --- p.68Chapter 3.5.1 --- Properties of Inverse Semigroup Algebras --- p.69Chapter 3.5.2 --- Radical of Algebras of Clifford Semigroups --- p.72Chapter 3.5.3 --- Semiprimitivity Problems of Inverse Semigroup Algebras --- p.73Chapter 3.6 --- Other Semigroup Algebras --- p.76Chapter 3.6.1 --- Completely Regular Semigroup Algebras --- p.76Chapter 3.6.2 --- Separative Semigroup Algebras --- p.77Chapter 3.7 --- Radicals of Pi-semigroup Algebras --- p.80Chapter 3.7.1 --- PI-Algebras --- p.80Chapter 3.7.2 --- Permutational Property and Algebras of Permutative Semigroups --- p.80Chapter 3.7.3 --- Radicals of PI-algebras --- p.82Chapter 4 --- Finiteness Conditions on Semigroup Algebras --- p.85Chapter 4.1 --- Introduction --- p.85Chapter 4.1.1 --- Preliminaries --- p.85Chapter 4.1.2 --- Semilattice Graded Rings --- p.86Chapter 4.1.3 --- Group Graded Rings --- p.88Chapter 4.1.4 --- Groupoid Graded Rings --- p.89Chapter 4.1.5 --- Semigroup Graded PI-Algebras --- p.91Chapter 4.1.6 --- Application to Semigroup Algebras --- p.92Chapter 4.2 --- Semiprime and Goldie Rings --- p.92Chapter 4.3 --- Noetherian Semigroup Algebras --- p.99Chapter 4.4 --- Descending Chain Conditions --- p.107Chapter 4.4.1 --- Artinian Semigroup Graded Rings --- p.107Chapter 4.4.2 --- Semilocal Semigroup Algebras --- p.109Chapter 5 --- Dimensions and Second Layer Condition on Semigroup Algebras --- p.119Chapter 5.1 --- Dimensions --- p.119Chapter 5.1.1 --- Gelfand-Kirillov Dimension --- p.119Chapter 5.1.2 --- Classical Krull and Krull Dimensions --- p.121Chapter 5.2 --- The Growth and the Rank of Semigroups --- p.123Chapter 5.3 --- Dimensions on Semigroup Algebras --- p.124Chapter 5.4 --- Second Layer Condition --- p.128Notations and Abbreviations --- p.132Bibliography --- p.13