Math 501. Abstract Algebra II

1. Module theory:
   1. Basic properties. Projectives and injectives. Characterizations and examples of projective modules. Examples of injective modules. Left exactness of Hom. Split exact sequences.
   2. Chain conditions. Jordan-Holder theorem for modules. Noetherian and artinian rings and modules. Idempotents and direct sum decompositions.
   3. Semisimple group algebras. Wedderburn's Theorem for semisimple and simple algebras.
   4. Tensor product, bilinear maps, right exactness, extension of coefficients and simple examples.
2. Modules over a commutative ring:
   1. Structure of finitely generated modules and submodules of free modules over a PID. Elementary divisors and invariant factors. Smith canonical form. Applications to abelian groups and matrices.
   2. Hilbert's basis theorem. Rings of fractions.

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Last modified June 7, 2004