Rings:
Definition, examples.
Quadratic integer rings.
Examples of non-commutative rings.
The Hamilton quaternions.
Polynomial rings.
Matrix rings.
Units, zero-divisors, nilpotents, idempotents.
Subrings, Ideals.
Maximal and prime Ideals.
Left, right and two-sided ideals;.
Operations with ideals.
The ideal generated by a set.
Quotient rings. Ring homomorphism.
The isomorphism theorems, applications.
Finitely generated ideals.
Rings of fractions.
Integral Domain:
The Chinese remainder theorem.
Divisibility in integral domains, greatest common divisor, least common multiple.
Euclidean domains.
The Euclidean algorithm.
Principal ideal domains.
Prime and irreducible elements in an integral domain.
Gauss lemma, irreducibility criteria for polynomials.
Unique factorization domains.
Finite fields.
Polynomials in several variables.
Symmetric polynomials.
The fundamental theorem of symmetric polynomials.
