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 (pab implies pa or pb).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 (Onetoone Function).A function φ from a set A is called onetoone 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 onetoone,then βα is onetoone.
3.If α and β are onto,then βα is onto.
4.If α is onetoone 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 (OneStep 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 (TwoStep 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 onetoone 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 2Cycles).Every permutation in S
n
,n ≥ 1,is a product of 2cycles.
Lemma.If ε = β
1
β
2
β
r
,where the β’s are 2cycles,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 2cycles,then every decomposition of α into a product of 2cycles must have an even number of 2cycles.
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 2cycles,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 2cycles is called an even permutation.A permutation that can be expressed as a product of an odd
number of 2cycles 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 onetoone 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 (OrbitStabilizer 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 ntuples 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 nto1 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 primepower 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 (Zerodivisors).A zerodivisor 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 zerodivisors.
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 (twosided) 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 onetoone 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).
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