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)|.

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