MA377 Rings and Modules
Lecturer: Christian Bohning
Term(s): Term 2
Status for Mathematics students: List A
Commitment: 30 lectures
Assessment: 85% by 3-hour examination 15% coursework
Formal registration prerequisites: None
Assumed knowledge: The ring theory part of the second year Maths core:
MA266 Multilinear Algebra
Jordan normal forms
Smith normal forms over integers
Classification of finitely generated abelian groups
MA268 Algebra 3
Rings
Domains (UFD, PID, ED)
Chinese remainder theorem
Gauss lemma
Eisenstein criterion
Useful background: Interest in Algebra and good working knowledge of Linear Algebra
Synergies: The following modules go well together with Rings and Modules:
MA3G6 Commutative Algebra
MA3E1 Groups and Representations
Leads to: The following modules have this module listed as assumed knowledge or useful background:
MA4J8 Commutative Algebra II
MA453 Lie Algebras
MA4M6 Category Theory
MA4H8 Ring Theory
Content: A ring is an important fundamental concept in algebra and includes integers, polynomials and matrices as some of the basic examples. Ring theory has applications in number theory and geometry. A module over a ring is a generalization of vector space over a field. The study of modules over a ring R provides us with an insight into the structure of R. In this module we shall develop ring and module theory leading to the fundamental theorems of Wedderburn and some of its applications.
Aims: To realise the importance of rings and modules as central objects in algebra and to study some applications.
Objectives: By the end of the course the student should understand:
The importance of a ring as a fundamental object in algebra
The concept of a module as a generalisation of a vector space and an Abelian group
Constructions such as direct sum, product and tensor product
Simple modules, Schur's lemma
Semisimple modules, artinian modules, their endomorphisms, examples
Radical, simple and semisimple artinian rings, examples
The Artin-Wedderburn theorem
The concept of central simple algebras, the theorems of Wedderburn and Frobenius
Books: Recommended Reading:
Abstract Algebra by David S. Dummit, Richard M. Foote, ISBN: 0471433349
Noncommutative Algebra (Graduate Texts in Mathematics) by Benson Farb, R. Keith Dennis, ISBN: 038794057X
Lecturer: Christian Bohning
Term(s): Term 2
Status for Mathematics students: List A
Commitment: 30 lectures
Assessment: 85% by 3-hour examination 15% coursework
Formal registration prerequisites: None
Assumed knowledge: The ring theory part of the second year Maths core:
MA266 Multilinear Algebra
Jordan normal forms
Smith normal forms over integers
Classification of finitely generated abelian groups
MA268 Algebra 3
Rings
Domains (UFD, PID, ED)
Chinese remainder theorem
Gauss lemma
Eisenstein criterion
Useful background: Interest in Algebra and good working knowledge of Linear Algebra
Synergies: The following modules go well together with Rings and Modules:
MA3G6 Commutative Algebra
MA3E1 Groups and Representations
Leads to: The following modules have this module listed as assumed knowledge or useful background:
MA4J8 Commutative Algebra II
MA453 Lie Algebras
MA4M6 Category Theory
MA4H8 Ring Theory
Content: A ring is an important fundamental concept in algebra and includes integers, polynomials and matrices as some of the basic examples. Ring theory has applications in number theory and geometry. A module over a ring is a generalization of vector space over a field. The study of modules over a ring R provides us with an insight into the structure of R. In this module we shall develop ring and module theory leading to the fundamental theorems of Wedderburn and some of its applications.
Aims: To realise the importance of rings and modules as central objects in algebra and to study some applications.
Objectives: By the end of the course the student should understand:
The importance of a ring as a fundamental object in algebra
The concept of a module as a generalisation of a vector space and an Abelian group
Constructions such as direct sum, product and tensor product
Simple modules, Schur's lemma
Semisimple modules, artinian modules, their endomorphisms, examples
Radical, simple and semisimple artinian rings, examples
The Artin-Wedderburn theorem
The concept of central simple algebras, the theorems of Wedderburn and Frobenius
Books: Recommended Reading:
Abstract Algebra by David S. Dummit, Richard M. Foote, ISBN: 0471433349
Noncommutative Algebra (Graduate Texts in Mathematics) by Benson Farb, R. Keith Dennis, ISBN: 038794057X