Introductory description

We aim to study noncommutative rings with chain conditions. A commutative integral domain has a (unique) field of fractions. What happens if we drop the commutativity axiom? Do we now obtain a division ring of fractions? If not always then when exactly? Do we need to differentiate between the left hand side and the right hand side of the ring? Also, does the theory extend meaningfully to rings such as rings of matrices which contain zero divisors? We shall give precise answers to all these questions.
Topics covered in pursuit of the above will include prime and semiprime rings, Artinian rings, composition series, the singular submodule, Ore’s theorem leading up to Goldie’s theorems and their applications.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Irreducible modules, completely reducible modules, chain conditions, composition series

‘Nil implies nilpotent’ theorems. Hopkins’s theorem that DCC implies ACC for rings

Uniform modules, Goldie dimension, complements, rings of quotients, Ore’s theorem, Goldie’s theorem, Artinian quotient rings, behaviour of prime ideals in rings of quotients

The Jacobson radical, fully bounded Noetherian rings, The
Jacobson conjecture

Learning outcomes

By the end of the module, students should be able to:

• The knowledge and understanding that a student will be expected to have upon completion, such as : ‘a theoretical knowledge of the principles and methods of archaeology’ or ‘a knowledge of the major types of chemical reaction and the main characteristics associated with them.’
• Communication (written, verbal, graphical...) Numeracy Use of information technology (e.g. WP, www, databases, spreadsheets, specialist packages) Ability to learn Others (e.g. teamwork)
• (c) Cognitive Skills For example: ability in critical analysis; the ability to formulate and test concepts and hypotheses.
• (d) Subject-Specific/Professional Skills For example: laboratory skills; scientific support writing; research skills and methods.

Indicative reading list

A.W.Chatters and C.R.Hajarnavis, Rings with chain conditions (QA251.5.C4)
K.R.Goodearl and R.B.Warfield,Jr., An introduction to noncommutative Noetherian Rings (QA251.5.G6)
J.C.McConnell and J.C.Robson, Noncommutative Noetherian rings (QA251.5M2)
N.H.McCoy, Rings and ideals (QA247.M33)
L.H.Rowen, Ring Theory (QA247.R68)

Subject specific skills

Ring theory applies among others to systems of numbers and matrices. Students get training in applying an axiom in set theory to deal with infinite sets which occur in algebra. The knowledge they acquire in MA377 is extended and deeper results on rings and modules are proved. They learn how to generalise the concept of fractions leading to acquaintance with modern results on rings of quotients.

Transferable skills

Students will acquire skills to solve complex problems in algebra. This will be valuable in careers such as teaching or research. They will have sufficient supporting knowledge to begin postgraduate work in algebra, geometry or representation theory.