Math 320 Theorems and Definitions 1 Theorem 0.1 (Division ...

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Math 320 Theorems and Definitions 1
Theorem 0.1 (Division Algorithm).Let a and b be integers with b > 0.Then there exist unique integers q and
r with the property that a = bq +r,where 0 ≤ r < b.
Definition (Greatest Common Divisor,Relatively Prime Integers).The greatest common divisor of two
nonzero integers a and b is the largest of all common divisors of a and b.We denote this integer by gcd(a,b).When
gcd(a,b) = 1,we say that a and b are relatively prime.
Theorem 0.2 (GCD Is a Linear Combination).For any nonzero integers a and b,there exist integers s and t
such that gcd(a,b) = as +bt.Moreover,gcd(a,b) is the smallest positive integer of the form as +bt.
Euclid’s Lemma (p|ab implies p|a or p|b).If p is a prime that divides ab,then p divides a or p divides b.
Theorem 0.3 (Fundamental Theorem of Arithmetic).Every integer greater than 1 is a prime or a product
of primes.The product is unique,except for the order in which the factors appear.Thus,if n = p
1
p
2
   p
r
and
n = q
1
q
2
   q
s
,where the p’s and q’s are primes,then r = s and,after renumbering the q’s,we have p
i
= q
i
for all i.
Theorem 0.a.Let a,b,n ∈ Z with n > 0.Then a mod n = b mod n if and only if n divides a −b.
Theorem 0.b.If a mod n = a

mod n = r
1
and b mod n = b

mod n = r
2
,then
1.(ab) mod n = (a

b

) mod n = (r
1
r
2
) mod n,and
2.(a +b) mod n = (a

+b

) mod n = (r
1
+r
2
) mod n.
Definition (Least Common Multiple).The least common multiple of two nonzero integers a and b is the
smallest positive integer that is a multiple of both a and b.We will denote this integer by lcm(a,b).
Definition (Equivalence Relation).An equivalence relation on a set S is a set R of ordered pairs of elements of
S such that
1.(a,a) ∈ R for all a ∈ S (reflexive property)
2.(a,b) ∈ R implies (b,a) ∈ R (symmetric property)
3.(a,b) ∈ R and (b,c) ∈ R imply (a,c) ∈ R.(transitive property)
Definition (Partition).A partition of a set S is a collection of nonempty disjoint subsets of S whose union is S.
Theorem 0.6 (Equivalence Classes Partition).The equivalence classes of an equivalence relation on a set S
constitute a partition of S.Conversely,for any partition P of S,there is an equivalence relation on S whose
equivalence classes are the elements of P.
Definition (Function,Mapping).A function (or mapping) φ from a set A to a set B is a rule that assigns to
each element a of A exactly one element b of B.The set A is called the domain of φ,and B is called the range of φ.
If φ assigns b to a,then b is called the image of a under φ.The subset of B comprising all the images of elements of
A is called the image of A under φ.
Definition (Composition of Functions).Let φ:A −→B and ψ:B −→C.The composition ψφ is the mapping
from A to C defined by (ψφ)(a) = ψ(φ(a)) for all a in A.
Definition (One-to-one Function).A function φ from a set A is called one-to-one if for every a
1
,a
2
∈ A,we
have φ(a
1
) = φ(a
2
) implies a
1
= a
2
.
Definition (Function from A onto B ).A function φ from a set A to a set B is said to be onto B if each
element of B is the image of at least one element of A.In symbols,φ:A −→B is onto if for each b ∈ B there is at
least one a ∈ A such that φ(a) = b.
Theorem 0.7 (Properties of Functions).Given functions α:A −→B,β:B −→C,and γ:C −→D.Then
1.γ(βα) = (γβ)α (associativity)
2.If α and β are one-to-one,then βα is one-to-one.
3.If α and β are onto,then βα is onto.
4.If α is one-to-one and onto,then there is a function α
−1
from B onto A such that (α
−1
α)(a) = a for all a ∈ A
and (αα
−1
)(b) = b for all b ∈ B.
Definition (Binary Operation).Let G be a set.A binary operation on G is a function that assigns each ordered
pair of elements of G exactly one element of G.
Math 320 Theorems and Definitions 2
Definition (Group).Let G be a nonempty set together with a binary operation (usually called multiplication)
that assigns to each ordered pair (a,b) of elements of G an element in G denoted by ab.We say that G is a group
under this operation if the following three properties are satisfied.
1.Associativity.The operation is associative;that is,(ab)c = a(bc) for all a,b,c ∈ G.
2.Identity.There is an element e (called the identity) in G such that ae = ea = a for all a ∈ G.
3.Inverses.For each element a in G,there is an element b in G (called an inverse of a) such that ab = ba = e.
Theorem 2.1 (Uniqueness of the Identity).In a group G,there is only one identity element.
Theorem 2.2 (Cancellation).In a group G,the right and left cancellation laws hold;that is,ba = ca implies
b = c and ab = ac implies b = c.
Theorem 2.3 (Uniqueness of Inverses).For each element a in a group G,there is a unique element b in G such
that ab = ba = e.
Definition (Order of a Group).The number of elements of a group (finite or infinite) is called its order.We
will use |G| to denote the order of G.
Definition (Order of an Element).The order of an element g in a group G is the smallest positive integer n
such that g
n
= e.(In additive notation,this would be ng = 0.) If no such integer exists,we see that g has infinite
order.The order of an element g is denoted by |g|.
Definition (Subgroup).If a subset H of a group G is itself a group under the operation of G,we say that H is a
subgroup of G.
Theorem 3.1 (One-Step Subgroup Test).Let G be a group and H a nonempty subset of G.Then H is a
subgroup of G if ab
−1
∈ H whenever a and b are in H.(In additive notation,H is a subgroup if a −b is in H
whenever a and b are in H.)
Theorem 3.2 (Two-Step Subgroup Test).Let G be a group and H a nonempty subset of G.Then H is a
subgroup of G if ab ∈ H whenever a,b ∈ H (closed under multiplication),and a
−1
∈ H whenever a ∈ H (closed
under taking inverses).
Theorem 3.3 (Finite Subgroup Test).Let H be a nonempty finite subset of a group G.Then H is a subgroup
of G if H is closed under the operation of G.
Theorem 3.4 (hai Is a Subgroup).Let G be a group,and let a be any element of G.Then hai = {a
n
:n ∈ Z}
is a subgroup of G.
Definition (Center of a Group).The center,Z(G),of a group G is the subset of elements in G that commute
with every element in G.In symbols,
Z(G) = {a ∈ G:ax = xa for all x ∈ G}.
Theorem 3.5 (Center Is a Subgroup).The center of a group G is a subgroup of G.
Definition (Centralizer of a in G ).Let a be a fixed element of a group G.The centralizer of a in G,C(a),is
the set of all elements that commute with a.In symbols,C(a) = {g ∈ G:ga = ag}.
Theorem 3.6 (C(a) is a subgroup).For each a ∈ G,the centralizer of a is a subgroup of G.
Theorem 4.1 (Criterion for a
i
= a
j
).Let G be a group,and let a ∈ G.If a has infinite order,then all distinct
powers of a are distinct group elements.If a has finite order,say n,then hai =
￿
e,a,a
2
,...,a
n−1
￿
and a
i
= a
j
if
and only if n divides i −j.
Corollary 1 (|a| = | hai |).For any group element a,we have |a| = | hai |.
Corollary 2 (a
k
= e Implies that |a| Divides k).Let G be a group and let a be an element of order n in G.If
a
k
= e,then n divides k.
Theorem 4.2 (
￿
a
k
￿
=
￿
a
gcd(n,k)
￿
).Let a be an element of order n in a group and let k be a positive integer.
Then
￿
a
k
￿
=
￿
a
gcd(n,k)
￿
and |a
k
| = n/gcd(n,k).
Corollary 1 (Criterion for
￿
a
i
￿
=
￿
a
j
￿
).Let |a| = n.Then
￿
a
i
￿
=
￿
a
j
￿
if and only if gcd(n,i) = gcd(n,j).
Corollary 2 (Generators of Cyclic Groups).Let G = hai be a cyclic group of order n.Then G =
￿
a
k
￿
if and
only if gcd(n,k) = 1.
Math 320 Theorems and Definitions 3
Corollary 3 (Generators of Z
n
).An integer k in Z
n
is a generator of Z
n
if and only if gcd(n,k) = 1.
Theorem 4.3 (Fundamental Theorem of Cyclic Groups).Every subgroup of a cyclic group is cyclic.
Moreover,if | hai | = n,then the order of any subgroup of hai is a divisor of n,and,for each positive divisor k of n,
the group hai has exactly one subgroup of order k,namely
￿
a
n/k
￿
.
Corollary (Subgroups of Z
n
).For each positive divisor k of n,the set hn/ki is the unique subgroup of Z
n
of
order k;moreover,these are the only subgroups of Z
n
.
Definition (Euler φ Function).For any integer n > 1,we define the Euler φ function as follows:φ(n) is the
number of positive integers less than n and relatively prime to n.
Theorem 4.4 (Number of Elements of Each Order in a Cyclic Group).If d is a positive divisor of n,the
number of elements of order d in a cyclic group of order n is φ(d).
Corollary (Number of Elements of Order d in a Finite Group).In a finite group the number of elements of
order d is divisible by φ(d).
Definitions (Permutation of A,Permutation Group of A).A permutation of a set A is a function from A to
A that is both one-to-one and onto.A permutation group of a set A is a set of permutations of A that forms a
group under function composition.
Theorem 5.1 (Products of Disjoint Cycles).Every permutation of a finite set can be written as a cycle or as
a product of disjoint cycles.
Theorem 5.2 (Disjoint Cycles Commute).If the pair of cycles α = (a
1
,a
2
,...,a
m
) and β = (b
1
,b
2
,...,b
n
)
have no entries in common,then αβ = βα.
Theorem 5.3 (Order of a Permutation).The order of a permutation of a finite set written in disjoint cycle
form is the least common multiple of the lengths of the cycles.
Theorem 5.4 (Product of 2-Cycles).Every permutation in S
n
,n ≥ 1,is a product of 2-cycles.
Lemma.If ε = β
1
β
2
   β
r
,where the β’s are 2-cycles,then r is even.
Theorem 5.5 (Always Even or Always Odd).If a permutation α can be expressed as a product of an even
number of 2-cycles,then every decomposition of α into a product of 2-cycles must have an even number of 2-cycles.
In symbols,if
α = β
1
β
2
   β
r
and α = γ
1
γ
2
   γ
s
,
where the β’s and the γ’s are 2-cycles,then r and s are both even or both odd.
Definition (Even and Odd Permutations).A permutation that can be expressed as a product of an even
number of 2-cycles is called an even permutation.A permutation that can be expressed as a product of an odd
number of 2-cycles is called an odd permutation.
Theorem 5.6 (Even Permutations Form a Group).The set of even permutations in S
n
forms a subgroup of
S
n
.
Definition (Alternating Group of Degree n).The group of even permutations of n symbols is denoted by A
n
and is called the alternating group of degree n.
Theorem 5.7.For n > 1,the group A
n
has order n!/2.
Definition (Group Isomorphism).An isomorphism φ from a group G to a group
G is a one-to-one mapping (or
function) from G onto
G that preserves the group operation.That is,
φ(ab) = φ(a)φ(b) for all a,b ∈ G.
If there is an isomorphism from G onto
G,we say that G and
G are isomorphic and write G ≈
G.
Theorem 6.1 (Cayley’s Theorem).Every group is isomorphic to a group of permutations.
Math 320 Theorems and Definitions 4
Theorem 6.2 (Properties of Isomorphisms Acting on Elements).Suppose that φ is an isomorphism from a
group G onto a group
G.Then
1.φ carries the identity of G to the identity of
G.
2.For every integer n and for every group element a ∈ G,φ(a
n
) = [φ(a)]
n
.
3.For any elements a and b in G,a and b commute if and only if φ(a) and φ(b) commute.
4.|a| = |φ(a)| for all a in G (isomorphisms preserve orders).
5.For a fixed integer k and a fixed group element b in G,the equation x
k
= b has the same number of solutions
in G as does the equation x
k
= φ(b) in
G.
Theorem 6.3 (Properties of Isomorphisms Acting on Groups).Suppose that φ is an isomorphism from a
group G onto a group
G.Then
1.G is Abelian if and only if
G is Abelian.
2.G is cyclic if and only if
G is cyclic.
3.φ
−1
is an isomorphism from
G onto G.
4.If K is a subgroup of G,then φ(K) = {φ(k):k ∈ K} is a subgroup of
G.
Definition (Automorphism).An isomorphism from a group G onto itself is called an automorphism of G.
Definition (Inner Automorphism Induced by a ).Let G be a group,and let a ∈ G.The function φ
a
defined
by φ
a
(x) = axa
−1
for all x ∈ G is called the inner automorphism of G induced by a.
Theorem 6.4 (AutG and Inn(G) Are Groups).The set of automorphisms of a group and the set of inner
automorphisms of a group are both groups under the operation of function composition.
Theorem 6.5 (AutZ
n
≈ U(n) ).For every positive integer n,AutZ
n
is isomorphic to U(n).
Definition (Coset of H in G ).Let G be a group and H ⊆ G.For any a ∈ G,the set {ah:h ∈ H} is denoted by
aH.Analagously,Ha = {ha:h ∈ H} and aHa
−1
=
￿
aha
−1
:h ∈ H
￿
.When H is a subgroup of G,the set aH is
called the left coset of H in G containing a,whereas Ha is called the right coset of H in G containing a.In this
case,the element a is called the coset representative of aH (or Ha).We use |aH| to denote the number of elements
in the set aH,and |Ha| to denote the number of elements in Ha.
Lemma (Properties of Cosets).Let H be a subgroup of G,and let a and b belong to G.Then
1.a ∈ aH,
2.aH = H if and only if a ∈ H,
3.aH = bH or aH ∩ bH = ∅,
4.aH = bH if and only if a
−1
b ∈ H,
5.|aH| = |bH|,(The number of elements in aH and in bH are equal.)
6.aH = Ha if and only if H = aHa
−1
,
7.aH is a subgroup of G if and only if a ∈ H.
Theorem 7.1 (Lagrange’s Theorem:|H| Divides |G| ).If G is a finite group and H is a subgroup of G,then
|H| divides |G|.Moreover,the number of distinct left (right) cosets of H in G is |G|/|H|.
Definition (Index of a Subgroup).Let H be a subgroup of a group G.The index of H in G is the number of
distinct left cosets of H in G.This number is denoted by |G:H|.
Corollary 1 (|G:H| = |G|/|H| ).If G is a finite group and H is a subgroup of G,then |G:H| = |G|/|H|.
Corollary 2 (|a| divides |G| ).In a finite group,the order of each element of the group divides the order of the
group.
Corollary 3 (Groups of Prime Order Are Cyclic ).A group of prime order is cyclic.
Corollary 4 (a
|G|
= e ).Let G be a finite group,and let a ∈ G.Then a
|G|
= e.
Corollary 5 (Fermat’s Little Theorem ).For every integer a and every prime p,a
p
mod p = a mod p.
Theorem 7.2 (Classification of Groups of Order 2p).Let G be a group of order 2p,where p is a prime
greater than 2.Then G is isomorphic to Z
2p
or to D
p
.
Math 320 Theorems and Definitions 5
Definition (Stabilizer of a Point).Let G be a group of permutations of a set S.For each i ∈ S,let
stab
G
(i) = {φ ∈ G:φ(i) = i}.
We call stab
G
(i) the stabilizer of i in G.
Definition (Orbit of a Point).Let G be a group of permutations of a set S.For each i ∈ S,let
orb
G
(i) = {φ(i):φ ∈ G}.
The set orb
G
(i) is a subset of S called the orbit of i under G.We use |orb
G
(i)| to denote the number of elements in
orb
G
(i).
Theorem 7.3 (Orbit-Stabilizer Theorem).Let G be a finite group of permutations of a set S.Then for any
i ∈ S,
|G| = |orb
G
(i)|  |stab
G
(i)|.
Definition (External Direct Product).Let G
1
,G
2
,...,G
n
be a finite collection of groups.The external direct
product of G
1
,G
2
,...,G
n
,written as G
1
⊕G
2
⊕   ⊕G
n
,is the set of all n-tuples for which the ith component is
an element of G
i
and the operation is componentwise.
Theorem 8.1 (Order of an Element in a Direct Product).The order of an element in a direct product of a
finite number of finite groups is the least common multiple of the orders of the components of the element.In
symbols,
|(g
1
,g
2
,...,g
n
)| = lcm(|g
1
|,|g
2
|,...,|g
n
|).
Theorem 8.2 (Criterion for G⊕H to Be Cyclic).Let G and H be finite cyclic groups.Then G⊕H is cyclic
if and only if |G| and |H| are relatively prime.
Corollary 1 (Criterion for G
1
⊕G
2
⊕   ⊕G
n
to Be Cyclic).An external direct product G
1
⊕G
2
⊕   ⊕G
n
of a finite number of cyclic groups is cyclic if and only if |G
i
| and |G
j
| are relatively prime when i 6= j.
Corollary 2 (Criterion for Z
n
1
n
2
∙∙∙n
k
≈ Z
n
1
⊕Z
n
2
⊕   ⊕Z
n
k
).Let m= n
1
n
2
   n
k
.Then Z
m
is isomorphic
to Z
n
1
⊕Z
n
2
⊕   ⊕Z
n
k
if and only if n
i
and n
j
are relatively prime when i 6= j.
Theorem 8.3 (U(n) as an External Direct Product).Suppose s and t are relatively prime.Then U(st) is
isomorphic to the external direct product of U(s) and U(t).In short,
U(st) ≈ U(s) ⊕U(t).
Moreover,U
s
(st) is isomorphic to U(t) and U
t
(st) is isomorphic to U(s).
Corollary.Let m= n
1
n
2
   n
k
,where gcd(n
i
,n
j
) = 1 for i 6= j.Then,
U(m) ≈ U(n
1
) ⊕U(n
2
) ⊕   ⊕U(n
k
).
Definition (Normal Subgroup).A subgroup H of a group G is called a normal subgroup of G if aH = Ha for
all a ∈ G.We denote this by H ⊳ G.
Theorem 9.1 (Normal Subgroup Test).A subgroup H of G is normal in G if and only if xHx
−1
⊆ H for all
x ∈ G.
Theorem 9.2 (Factor Groups).Let G be a group and H a normal subgroup of G.The set G/H = {aH:a ∈ G}
is a group under the operation (aH)(bH) = abH.
Theorem 9.3 (The G/Z Theorem).Let G be a group,and let Z(G) be the center of G.If G/Z(G) is cyclic,
then G is Abelian.
Theorem 9.4 (G/Z(G) ≈ Inn(G) ).For any group G,G/Z(G) is isomorphic to Inn(G).
Theorem 9.5 (Cauchy’s Theorem for Abelian Groups).Let G be a finite Abelian group and let p be a
prime that divides the order of G.Then G has an element of order p.
Definition (Internal Direct Product of H and K ).We say that G is the internal direct product of H and K
and write G = H ×K if H and K are normal subgroups of G and
G = HK and H ∩ K = {e}.
Math 320 Theorems and Definitions 6
Definition (Internal Direct Product H
1
×H
2
×   ×H
n
).Let H
1
,H
2
,...,H
n
be a finite collection of
normal subgroups of G.We say that G is the internal direct product of H
1
,H
2
,...,H
n
and write
G = H
1
×H
2
×   ×H
n
if
1.G = H
1
H
2
   H
n
= {h
1
h
2
   h
n
:h
i
∈ H
i
}
2.(H
1
H
2
   H
i
) ∩ H
i+1
= {e} for i = 1,2,...,n −1.
Theorem 9.6 (H
1
×H
2
×   ×H
n
≈ H
1
⊕H
2
⊕   ⊕H
n
).If a group G is the internal direct product of a
finite number of subgroups H
1
,H
2
,...,H
n
,then G is isomorphic to the external direct product of H
1
,H
2
,...,H
n
.
Definition (Group Homomorphism).A homomorphism φ from a group G to a group
G is a mapping from G
into
G that preserves the group operation;that is,φ(ab) = φ(a)φ(b) for all a,b ∈ G.
Definition (Kernel of a Homomorphism).The kernel of a homomorphism φ from a group G to a group with
identity e is the set {x ∈ G:φ(x) = e}.The kernel of φ is denoted by Ker φ.
Theorem 10.1 (Properties of Elements under Homomorphisms).Let φ be a homomorphism from a group
G to a group
G and let g be an element of G.
1.φ carries the identity of G to the identity of
G.
2.φ(g
n
) = (φ(g))
n
for all n ∈ Z.
3.If |g| is finite,then |φ(g)| divides |g|.
4.Ker φ is a subgroup of G.
5.If φ(g) = g

,then φ
−1
(g

) = {x ∈ G:φ(x) = g

} = gKer φ.
Theorem 10.2 (Properties of Subgroups under Homomorphisms).Let φ be a homomorphism from a
group G to a group
G and let H be a subgroup of G.Then
1.φ(H) = {φ(h):h ∈ H} is a subgroup of
G.
2.If H is cyclic,then φ(H) is cyclic.
3.If H is Abelian,then φ(H) is Abelian.
4.If H is normal in G,then φ(H) is normal in φ(G).
5.If |Ker φ| = n,then φ is an n-to-1 mapping from G onto φ(G).
6.If |H| = n,then |φ(H)| divides n.
7.If
K is a subgroup of
G,then φ
−1
(
K) =
￿
k ∈ G:φ(k) ∈
K
￿
is a subgroup of G.
8.If
K is a normal subgroup of
G,then φ
−1
(
K) =
￿
k ∈ G:φ(k) ∈
K
￿
is a normal subgroup of G.
9.If φ is onto and Ker φ = {e},then φ is an isomorphism from G to
G.
Corollary (Kernels Are Normal).Let φ be a group homomorphism from G to
G.Then Ker φ is a normal
subgroup of G.
Theorem 10.3 (First Isomorphism Theorem).Let φ be a group homomorphism from G to
G.Then the
mapping from G/Ker φ to φ(G),given by gKer φ → φ(g),is an isomorphism.In symbols,G/Ker φ ≈ φ(G).
Theorem 10.4 (Normal Subgroups Are Kernels).Every normal subgroup of a group G is the kernel of a
homomorphism of G.In particular,a normal subgroup N is the kernel of the mapping g →gN from G to G/N.
Theorem 11.1 (Fundamental Theorem of Finite Abelian Groups).Every finite Abelian group is a direct
product of cyclic groups of prime-power order.Moreover,the number of terms in the product and the orders of the
cyclic groups are uniquely determined by the group.
Definition (Ring).A ring R is a nonempty set with two binary operations,addition (denoted by a +b) and
multiplication (denoted by ab),such that for all a,b,c in R:
1.a +b = b +a.
2.(a +b) +c = a +(b +c).
3.There is an additive identity 0.That is,there is an element 0 in R such that a +0 = a for all a ∈ R.
4.There is an element −a in R such that a +(−a) = 0.
5.a(bc) = (ab)c.
6.a(b +c) = ab +ac and (b +c)a = ba +ca.
Math 320 Theorems and Definitions 7
Theorem 12.1 (Rules of Multiplication).Let a,b,and c belong to a ring R.Then
1.a0 = 0a = 0.
2.a(−b) = (−a)b = −(ab).
3.(−a)(−b) = ab.
4.a(b −c) = ab −ac and (b −c)a = ba −ca.
Furthermore,if R has a unity element 1,then
5.(−1)a = −a.
6.(−1)(−1) = 1.
Theorem 12.2 (Uniqueness of the Unity and Inverses).If a ring has a unity,it is unique.If a ring element
has a multiplicative inverse,it is unique.
Definition (Subring).A subset S of a ring R is a subring of R if S is itself a ring with the operations of R.
Theorem 12.3 (Subring Test).A nonempty subset S of a ring R is a subring if S is closed under subtraction
and multiplication;that is,if a −b and ab are in S whenever a and b are in S.
Definition (Zero-divisors).A zero-divisor is a nonzero element a of a commutative ring R such that there is a
nonzero element b ∈ R with ab = 0.
Definition (Integral Domain).An integral domain is a commutative ring with unity and no zero-divisors.
Theorem 13.1 (Cancellation).Let a,b,and c belong to an integral domain.If a 6= 0 and ab = ac,then b = c.
Definition (Field).A field is a commutative ring with unity in which every nonzero element is a unit.
Theorem 13.2 (Finite Integral Domains Are Fields).A finite integral domain is a field.
Corollary (Z
p
Is a Field).For every prime p,Z
p
,the ring of integers modulo p,is a field.
Definition (Characteristic of a Ring).The characteristic of a ring R is the least positive integer n such that
nx = 0 for all x ∈ R.If no such integer exists,we say that R has characteristic 0.The characteristic of R is denoted
by char R.
Theorem 13.3 (Characteristic of a Ring with Unity).Let R be a ring with unity 1.If 1 has infinite order
under addition,then the characteristic of R is 0.If 1 has order n under addition,then the characteristic of R is n.
Theorem 13.4 (Characteristic of an Integral Domain).The characteristic of an integral domain is 0 or
prime.
Definition (Ideal).A subring A of a ring R is called a (two-sided) ideal of R if for every r ∈ R and every a ∈ A
both ra and ar are in A.
Theorem 14.1 (Ideal Test).A nonempty subset A of a ring R is an ideal of R if
1.a −b ∈ A whenever a,b ∈ A
2.ra and ar are in A whenever a ∈ A and r ∈ R.
Theorem 14.2 (Existence of Factor Rings).Let R be a ring and let A be a subring of R.The set of cosets
{r +A:r ∈ R} is a ring under the operations
(s +A) + (t +a) = s +t + A and (s +A)(t +A) = st + A
if and only if A is an ideal of R.
Definition (Prime Ideal,Maximal Ideal).A prime ideal A of a commutative ring R is a proper ideal of R such
that a,b ∈ R and ab ∈ A implies a ∈ A or b ∈ A.A proper ideal A of a commutative ring R is a maximal ideal of R
if,whenever B is an ideal of R and A ⊆ B ⊆ R,then B = A or B = R.
Theorem 14.3 (R/A Is an Integral Domain if and Only if A Is Prime).Let R be a commutative ring with
unity and let A be an ideal of R.Then R/A is an integral domain if and only if A is prime.
Theorem 14.4 (R/A Is a Field if and Only if A is Maximal).Let R be a commutative ring with unity and
let A be an ideal of R.Then R/A is a field if and only if A is maximal.
Math 320 Theorems and Definitions 8
Definition (Ring Homomorphism,Ring Isomorphism).A ring homomorphism φ from a ring R to a ring S
is a mapping from R to S that preserves the two ring operations;that is,for all a,b in R,
φ(a +b) = φ(a) + φ(b) and φ(ab) = φ(a)φ(b).
A ring homomorphism that is both one-to-one and onto is called a ring isomorphism.
Theorem 15.1 (Properties of Ring Homomorphisms).Let φ:R −→S be a ring homomorphism,and let A
be a subring of R and B be an ideal of S.
1.For any r ∈ R and any positive integer n,we have φ(nr) = nφ(r) and φ(r
n
) = (φ(r))
n
.
2.φ(A) = {φ(a):a ∈ A} is a subring of S.
3.If A is an ideal and φ maps onto S,then φ(A) is an ideal.
4.φ
−1
(B) = {r ∈ R:φ(r) ∈ B} is an ideal of R.
5.If R is commutative,then φ(R) is commutative.
6.If R has a unity 1,S 6= ∅,and φ is onto,then φ(1) is the unity of S.
7.φ is an isomorphism if and only if φ is onto and Ker φ = {0}.
8.If φ is an isomorphism from R onto S,then φ
−1
is an isomorphism from S onto R.
Theorem 15.2 (Kernels are Ideals).Let φ:R −→S be a ring homomorphism.Then Ker φ is an ideal of R.
Theorem 15.3 (First Isomorphism Theorem for Rings).Let φ:R −→S be a ring homomorphism.Then the
mapping from R/Ker φ to φ(R),given by r +Ker φ → φ(r),is an isomorphism.In symbols,R/Ker φ ≈ φ(R).
Theorem 15.4 (Ideals are Kernels).Every ideal of a ring R is the kernel of a ring homomorphism of R.In
particular,an ideal A is the kernel of the mapping r → r +A from R to R/A.
Definition (Elements Fixed by a Permutation).Let G be a group of permutations of a set S.For each φ ∈ G,
let
fix(φ) = {i ∈ S:φ(i) = i}.
This set is called the elements fixed by φ.
Theorem 29.1 (Burnside’s Theorem).Let G be a finite group of permutations on a set S.Then the number of
orbits of G on S is given by
1
|G|
￿
φ∈G
|fix(φ)|.