Continuation of Intro to rings, modules, ideals
October 20
This material is covered in sections 10.1-10.4
1
Congruence
Let n = 0 Z and usuallywe will be thinking of this n as a positive number. For x, y Z we
say
x y mod n
if equivalently:
y x is divisibl
Finitely generated abelian groups
December 1
1
Rings satisfying the Divisible Diagonalizable Hypothesis
Divisible Diagonalizable Hypothesis (DDH): Let R be an integral domain for which every
m n matrix over R is divisibly diagonalizable (i.e., diagonaliza
Homework Set due Nov.5, and Sylow Theory
October 29
Recall: The Sylow subgroup material is covered in section 6.4
1
More on idempotents
Exercise 1 Let R be the set of continuous real-valued functions on the set Sconsisting of all nonzero
real numbers of a
Jordan Canonical Form and other applications of our theory of modules
December 3
(Material from Chapter 12 section 7 of Artins Algebra
1
Jordan canonical Form
Let k be a eld. Let V := k[T ]/p(T ) be a cyclic k[T ]-module where we assume that p(T ) is a
(m
Intro to the Jordan-Hlder Theorem and Homework set due Nov 19
o
November 12
Material covered, for example, in Dummit and Foote Abstract Algebra pp. 102-106.
1
Series or chains of groups
If G is a group, well call a series (or synonymously a chain of subgr
More on Sylow Theory
November 3
Recall: The Sylow subgroup material is covered in section 6.4
1
Whirlwind Sylow
Theorem 1 Fix G a nite group and p a prime number. Write |G| = pe m with (p, m) = 1.
The number of p-Sylow subgroups in G is congruent to 1 mo
More modules and presentation matrices
November 24
Material covered in Artins Chapter 12 section 1-6; and also more generally covered
in Dummit and Foote Abstract Algebra Chapter 12
1
Standard Row and Column Operations
Recall that our rings are commutativ
Modules and presentation matrices
November 19
Material covered in Artin Chapter 12 section 1-6.
1
Recall
that our rings are commutative with unit, and were talking about modules over such rings. Recall
what it means for a set of elements cfw_x1 , x2 , . .
Continuation of Intro to rings, modules, and Homework due October 29
October 22
This material is covered in sections 10.1-10.4
1
Recalling modules
More examples, the category of R-modules.
Exercise 1
1. Let R be a commutative ring with unit and M, N two R
Modules
November 17
Material covered in Artin Chapter 12 section 1-6.
1
Recall modules; set up conventions; basic vocabulary
We will work with commutative rings with unit R. The category of R-modules. If U, V are Rmodules we have the direct sum U V as an
p-primary components and Sylow subgroups
October 27
The Sylow subgroup material is covered in section 6.4
1
Abelian groups, alias Z-modules
Denition 1 Let p be a prime number. A nite group G is called a p-group it is has order a
power of p.
Form now on in