1. Group theory:
The theory of groups is one of the oldest and richest branches of algebra.
(1) Explanation of fundamental concepts:
Permutation groups, Cyclic groups, Group homomorphisms, Quotient groups, Group actions;
(2) Finite groups being the basis of Galois' discoveries in the theory of equations:
Sylow theorems, Classification of finite groups, Simple groups, Solvable groups.

2. Ring theory:
The theory of rings grew out of the study of two particular classes of rings, polynomial rings and the "integers" of an algebraic number field.
(1) Basic concepts concerning rings:
Ideals, Rings of fractions, Euclidean domains, Principal ideal domains, Unique factorization domains;
(2) Standard examples:
Matrix rings, Polynomial rings, Rings of quadratic algebraic integers;
(3) Computational techniques enhancing the development:
Resultant,Grobner basis.