We continue the studies of last semester. This semester,
we treat the basic theory of rings and modules.
We begin with a short discussion on category theory, as it
will put many things in context. We then treat the topics from Chapter 7
omitted from the previous courses, including free modules, direct sums,
chain conditions, and rings and modules of fractions. We conclude this section
with a proof of the Fundamental Theorem of Finitely Generated Modules over
a PID, modulo the proof already given in Math 524 for torsion modules.
Then we study Chapter 9, which treats a
variety of topics in ring and module theory, including Hilbert's
Nullstellensatz, tensor products and projectcive modules.
We conclude with Chapter 12, which treats semisimple
rings (with applications to representation theory) and Dedekind domains
(with applications to algebraic number theory).
Your grade will be based entirely on take home problem sets.