31
Prime elements
31.1 Denition. Let R be an integral domain, and let a, b R. We say that a
divides b if b = ac for some c R. We then write: a | b.
31.2 Proposition. If R is an integral domain and a, b R then a b i a | b
and b | a.
Proof. Exercise.
31.3 D
13
More on free abelian groups
Recall. G is a free abelian group if
G
=
for some set I.
Z
iI
13.1 Denition. Let G be an abelian group. A set B G is a basis of G if
B generates G
if for some x1 , . . . xk B and n1 , . . . , nk Z we have
n1 x1 + + nk xk =
7
Index of a subgroup and order of an element
7.1 Denition. Let H be a subgroup of G. Then
[G : H] := the number of distinct left cosets of H in G
This number is called the index of H in G.
7.2 Note.
1) The number of left cosets of H in G is the same as t
15
Permutation representations and G-sets
Recall. If C is a category and c C then
Aut(c) = the group of automorphisms of c
15.1 Denition. A representation of a group G in a category C is a homomorphism
: G Aut(c)
Special types of representations:
linear
24
Nilpotent groups
24.1. Recall that if G is a group then
Z(G) = cfw_a G | ab = ba for all b G
Note that Z(G) G. Take the canonical epimorphism : G G/Z(G). Since
Z (G/Z(G) G/Z(G) we have:
1 (Z (G/Z(G)
G
Dene:
Z1 (G) :=Z(G)
1
Zi (G) :=i (Z (G/Zi1 (G)
whe
36
Rings of fractions
Recall. If R is a PID then R is a UFD.
In particular
Z is a UFD
if F is a eld then F[x] is a UFD.
Goal. If R is a UFD then so is R[x].
Idea of proof.
1) Find an embedding R F where F is a eld.
2) If p(x) R[x] then p(x) F[x] and sin
43
Projective modules
43.1 Note. If F is a free R-module and P F is a submodule then P need
not be free even if P is a direct summand of F .
Take e.g. R = Z/6Z. Notice that Z/2Z and Z/3Z are Z/6Z-modules and we
have an isomorphism of Z/6Z-modules:
Z/6Z Z/
21
Symmetric and alternating groups
Recall. The symmetric group on n letters is the group
Sn = Perm(cfw_1, . . . , n)
21.1 Theorem (Cayley). If G is a group of order n then G is isomorphic to a
subgroup of Sn .
Proof. Let S be the set of all elements of G
9
Direct products, direct sums, and free abelian
groups
9.1 Denition. A direct product of family of groups cfw_Gi iI is a group
a
G
iI Gi dened as follows. As a set
iI i is the cartesian product of the
groups Gi . Given elements (ai )iI , (bi )iI iI Gi we
46
Injective modules
Recall. If R is a ring with identity then an R-module P is projective i one of
the following equivalent conditions holds:
1) For any homomorphism f : P N and an epimorphism g : M N there
is a homomorphism h : P M such that the followi
18
Application: groups of order pq
Recall. If G is a group, |G| = p where p is a prime then G Z/pZ.
=
Goal. Classify all groups of order pq where p, q are prime numbers.
18.1 Proposition. If G is a group of order p2 for some prime p then either
G Z/p2 Z o
Note. From now on all rings are commutative rings with identity 1 = 0
unless stated otherwise.
27
Principal ideal domains and Euclidean rings
27.1 Denition. If R is a ring and S is a subset of R then denote
S = the smallest ideal of R that contains S
We s
1
Monoids and groups
1.1 Denition. A monoid is a set M together with a map
M M M,
(x, y) x y
such that
(i) (x y) z = x (y z) x, y, z M (associativity);
(ii) e M such that
xe=ex=x
for all x M (e = the identity element of M ).
1.2 Examples.
1) Z with additi
38
Irreducibility criteria in rings of polynomials
38.1 Theorem. Let p(x), q(x) R[x] be polynomials such that
p(x) = a0 + a1 x + . . . + an xn ,
q(x) = b0 + b1 x + . . . + bm xm
and an , bm = 0. If bm is a unit in R then there exist unique polynomials
r(x
42
Invariant basis number
42.1 Denition. A ring R has the invariant basis number (IBN) property if for
any free R-module F and for any bases two B, B of F we have |B| = |B |.
42.2 Denition. If a ring R has IBN then for a free R-module F the rank of
F is t