Topics in groups theory 12 hours
Subgroups of finite p-groups. Sylow's theorem. Semidirect product of groups.

Rings 30 hours
Definitions. Zero divisors. Invertible elements. Integral rings and integral domains. Division rings. Fields. Quaternions. Field of fractions of a domain.
Subrings, ring homomorphisms. Isomorphism theorem. Ideal and quotient ring. Characteristic of a ring. Prime subfield of a ring.
Polynomial rings. Formal power series ring. Universal property of polynomial rings. Multivariate polynomials.
Divisibility in integral domain. Associate elements. Prime and irreducible elements. Unique factorization domains (UFD) and their characterization. Highest common factor and least common multiple. Euclidean domains. Euclid's algorithm. Division of polynomials. Zeroes of polynomials. Factorization of polynomials. Gauss' lemma. Theorem: the polynomial ring over a UFD is a UFD.
Extensions of fields. Algebraic elements. Minimal polynomial. Transcendent elements. Roots of unity. Finite fields.

Modules 22 hours
Endomorphism ring of an abelian group. Left and right modules over a ring. Opposite ring. Left and right ideals. Submodules. Module homomorphisms. Direct product of modules. Independent modules. Linear independent elements. Submodule generated by a subset. Free modules. Basis. Annihilators of elements and modules. Cyclic modules.
Principal ideal domains (PID). Classification of finitely generated modules over PID. Finitely generated abelian groups. Jordan's canonical form of a matrix. Minimum polynomial of a matrix.