Week 1, January 13, 2010

1/13: 
Review of principal ideal domains and unique factorization in
principal ideal domains.  
Definition of maximal ideals in a commutative ring.
Definition of prime ideals in a commutative ring.

1/15:
Characterization of maximal ideals: quotient ring is a field.
Every maximal ideal is a prime ideal.


Week 2, January 18, 2010

1/20: 
Some examples of commutative rings:
polynomials rings of the form R[x], R[x,y],
where R is the ring of all integers 
or a field such as rational numbers, real
numbers or complex numbers


1/22: 
How to think about factorization in an integral domain,
especially a PID, in terms of ideals.
Formulation of concepts such as gcd and lcm
in terms of ideals.
Formulation of Unique Factorization in terms of
ideals.
Every non-zero prime ideal in a PID is maximal.

Week 3, January 25, 2010

1/25:
Example of an integral domain which is not a UFD (hence not a PID)
Examples of factorization in the ring of Gaussian integers.


1/27:
Definition of Euclidean domains.
Every Euclidean domain is a PID.
The ring of Gaussian integers is a Euclidean domain,
hence a PID. 


1/29: 
Fraction field of an integral domain.
Examples of fractin fields.
Unique factorization domains.


Week 4, February 1, 2010

2/01:
Zorn's lemma.
Existence of maximal ideals containing a 
given proper ideal.

2/03
Beginning of the theorem:
If R is a UFD, then R[x] is a UFD.
Gauss lemma.
irreducible elements in R[x].

2/05
Finish the proof using Gauss lemma.


Week 5, February 8, 2010

2/08
Definition of modules.
Examples: \
modules over fields are vector spaces,
modules over integers are belian groups, 
a module over a polynomial F[x] over a field
corresponds to a linear operator on a vector space,
a module over a polynomial ring F[x,y] over a field
corresponds to two commuting linear operators.

2/12
Example of modules over matrix rings.
Definition of submodules, 
homomorphisms between module,
kernel and image of modules,
simple modules.


Week 6, February 15, 2010

2/15
An R-module M corresponds to a ring homomorphism
from R to End_{grp}(M)
A module over a group ring F[G] corresponds
to an F-linear representation of G,
or equivalently a group homomophism from G to GL(n).
Back to linear operators and canonical forms.

2/17
Beginning of structure theorem of finitely generated
modules over a PID. Statement of theorem.
Case for the ring of all integers.

2/19
Translation of the structure theorem into the standard
statement of rational canonical forms.
Equivalent statement of the structure theorem:
For every mxn matrix A with coefficients in a PID,
there exist an invertible mxm matrix C and an invertible
n\times n matrix D such that CAD is "diagonal".

2/22
Finite generation of submodules of a finitely generated module
over a PID.

2/24
Necessary and sufficient condition for a row (column) vector
to be a row (column) of an invertible matrix with coefficients
in a PID.