Topics in Ring Theory and Modules

About this course:

Ring theory is a branch of abstract algebra that deals with sets—for example, the collection of all integers—that admit both additive and multiplicative operations.  Modules generalize the notion of vector spaces, but with scalars drawn from a ring rather than a field.  Beginning with a survey of the basic notions of rings and ideals, the course explores some of the elegant algebraic structuring that defines the behavior of rings—both commutative and non-commutative—and their associated modules.  Topics to be discussed include the isomorphism theorems; ascending and descending chain conditions;  prime and maximal ideals; free, simple, and semi-simple modules; the Jacobson radical; and the Wedderburn-Artin Theorem.  Theory will be motivated by numerous examples drawn from familiar realms of number theory, linear algebra, and real analysis.