This is a standard course for beginners. We will consider a lot of examples.
Group Theory: Basic concepts, isomorphism theorems,
normal subgroups, Sylow theorems, direct products and free products of groups.
Groups acting on sets: orbits, cosets, stabilizers.
Alternating and Symmetric groups.
Basic Ring Theory: Fields, Principal Ideal Domains (PIDs),
matrix rings, division algebras, field of fractions.
Modules over a PID: Fundamental Theorem for abelian groups,
application to linear algebra: rational and Jordan canonical form.
Bilinear Forms: Alternating and symmetric forms, determinants.
Spectral theorem for normal matrices, classification over R and C.
Modules: Artinian and Noetherian modules.
Krull-Schmidt Theorem for modules of finite length.
Simple modules and Schur's Lemma, semisimple modules.
Finite-dimensional algebras: Simple and semisimple
algebras, Artin-Wedderburn Theorem, group rings, Maschke's Theorem.