MATH4RM Rings and Modules
Ring theory is one of the two main topics studied by algebraists (the other being group theory). The theory of rings and their modules is an active research area: in its regular listing of new research articles from around the world, the American Mathematical Society identified more than 70 items in ring theory for a three week long period in July 2012. It originally developed as a collection of (at the time brilliant) insights for proving and improving a wide range of theorems in algebra, by extracting their essence and discarding their superfluous hypotheses.
In linear algebra, a vector space is endowed with a scalar multiplication, these scalars coming from a field (often the complex field C), subject to certain rules. A module over a ring R is just a vector space where the field of scalars is replaced by the set of elements in R. Much of the theory of modules is concerned with extending the properties of vector spaces to modules. However, module theory can be much more complicated than that of vector spaces. For example, every vector space has a basis and the cardinalities of any two of its bases are equal. However, as seen in the paper, a module may not have a basis and, even if it does, it may have several bases of different cardinalities.
The interplay between rings and their modules is investigated. At times we examine a ring by considering a given set of its modules and at other times we consider a ring of a certain type to see how it influences the structure of its modules. The main goal is the Wedderburn-Artin Theorem, characterising semisimple Artinian rings. Such a ring can be regarded as the Sun in the solar system of ring theory and much of the theory involves measuring how far a ring is from this Sun. Our approach to the theorem uses projective and injective modules, the elements of what is known as homological algebra.
MATH342 Modern Algebra
John Clark (Room 216, phone 479 7781, email: email@example.com)
4 written assignments
There will be a three-hour exam.