Ring Homomorphisms
Definition: Ring Homomorphism, Ring
isomorphism
A ring homomorphism 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 R,
a + b = a + b and ab = ab.
A ring homomorphism that
Cosets and Lagranges Theorem
Definition: Coset of H in G
Let G be a group with subset H. For any
a G, the set ah / h H is denoted by aH.
Analogously, 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
Group Homomorphisms
Definition: Group homomorphism
A homomorphism from a group G to a
group G is a mapping from G into G such that
preserves the group operation; that is,
ab = ab.
Definition: Kernel of a homomorphism
The kernel of a homomorphism from a
g
Isomorphisms
Definition: Group Isomorphisms.
An isomorphism from G to G is a
one-to-one mapping (or function) from G onto
G that preserves the group operation. That is,
ab = ab for all a, b in G.
If there is an isomorphism from G onto G, we
say that G and
Introduction to rings
Definition: Ring
A ring R is a set with two binary operations,
addition (denoted by a + b) and multiplication
(denoted by ab), such that for all a, b, c R :
1. a + b = b + a.
2. a + b + c = a + b + c.
3. There is an additive identity
Finite Groups and Subgroups
Definition: The order of a Group
The number of elements in a group (finite or
infinite) is called the 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 t
Euclidean Algorithm:
For any pair of positive integers a and b, we
may find gcda, b by repeated use of division
to produce a decreasing sequence of integers
r 1 > r 2 >. . . . . . as follows:
a = bq 1 + r 1
0 < r1 < b
b = r1q2 + r2
0 < r2 < r1
r1 = r2q3 +
Permutation Groups
Definition: Permutation 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 on A that forms a group under
function composition.
Examples:
1.
Equivalence Relations:
Definition: An equivalence relation on a set S
is a set R of ordered pairs of elements of S
such that:
a, a R for all a S (reflexive property)
a, b R implies that b, a R (symmetric
property)
a, b R and b, c R imply that
a, c R (t
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. This
integer is denoted by lcma, b.
Modular Arithmetic
Definition: Let a and b be integers suc
Groups
Binary operation:
Definition: Let G be a set. A binary operation
on G is a function that assigns each ordered
pair of elements of G an element of G.
Definition: Group
Let G be a set together with a binary
operation that assigns to each ordered pair
Properties of Ring Homomorphisms: (Chapter 15, p283)
Proof of Properties:
Observe the following: Let r 2 R: Since the rings R and S are Abelian groups under
their respective additive operations, we have that maps the additive identity in R to the
additive
Introduction to Groups
Symmetries of a square:
Rotation of 0 0 (no change of position):
R0 =
1234
1234
Rotation of 90 0 (counterclockwise):
R 90 =
1234
2341
Rotation of 180 0 :
R 180 =
1234
3412
Rotation of 270 0 :
R 270 =
1234
4123
Flip about the horizon
External Direct Products
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
Well ordering principle:
Every nonempty set of positive integers
contains a smallest member.
Theorem 0.1: Division Algorithm
Let a and b be integers such that b > 0. Then
there exists unique integers q and r such that
a = bq + r, where 0 r < b.
Greatest c
Theorem 4.1: Criterion for a i = a j .
Let G be a group, and let a belong to G. If a
has infinite order, then a i = a j if and only if
i = j. If a has finite order, say n, then
a = e, a, a 2 , a 3 , . . . . . , a n1 and a i = a j if and
only if n divides
More Examples:
1. Is the set of irrational numbers a group
under multiplication?
2. Is the set U5 a group under
multiplication modulo 5?
3. Is the set of integers a group under the
subtraction operation?
Elementary Properties of Groups:
Theorem 2.1: In a
Integral Domains
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-divi
Corollary 1 Let be a homomorphism from G to G: Then Ker is a normal subgroup of
G:
Proof. We will rst show that Ker is a subgroup of G: Let a; b 2 Ker and suppose that e
is the identity of G: Now, by the 1-Step Subgroup Test, we need to check that ab1 2 K
Ideals and Factor Rings
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. That is: rA = ra /
a A A and Ar = ra / a A A for all
r R.
Theorem: Ideal Test
A nonempty subset A
Normal Subgroups and Factor Groups
Definition: Normal subgroup
A subgroup H of a group G is called a normal
subgroup of G if aH = Ha for all a in G. We
denote this by H G.
Theorem: Normal Subgroup Test
A subgroup H of G is normal in G if and only if
xHx 1
Cyclic Groups
Definition: Cyclic Group
We say that a group G is cyclic when there
exists an a in G such that G = a . Recall that
a = a n / n Z. We will refer to a as a
generator of G.
Examples:
1. Is Z a cyclic group under addition? If so,
find all the ge
Mathematical induction
Theorem 0.4: First Principle of
mathematical induction
Let S be a set of integers containing a.
Suppose that S has the property that
whenever some integer n a belongs to S,
then the integer n + 1 also belongs to S. Then,
S contains
Theorem 1 A finite integral domain is a field.
Proof. Let D be a finite integral domain with unity 1. Let a be any non-zero element of
D. If a = 1 then a is its own multiplicative inverse. We may assume that a = 1. Consider
the sequence a, a2 , a3 , . Sin
SOLUTIONS TO THE FINAL
(CHAPTERS 7-10)
MATH 141 FALL 2015 KUNIYUKI
250 POINTS TOTAL
1) Find the intersection point(s) of the graphs of 2x y = 0 and 12x 2 y 2 = 16 in
the usual xy-plane by solving a system, as we have done in class. Do not rely on
graphing
MIDTERM 4 SOLUTIONS
(CHAPTERS 5 AND 6: ANALYTIC & MISC. TRIGONOMETRY)
MATH 141 FALL 2015 KUNIYUKI
150 POINTS TOTAL: 47 FOR PART 1, AND 103 FOR PART 2
PART 1: USING SCIENTIFIC CALCULATORS (47 PTS.)
1) Find the length of Side a for the triangle below using
MIDTERM 3 SOLUTIONS
(CHAPTER 4)
INTRODUCTION TO TRIGONOMETRY; MATH 141 FALL 2015 KUNIYUKI
150 POINTS TOTAL: 30 FOR PART 1, AND 120 FOR PART 2
PART 1: USING SCIENTIFIC CALCULATORS (30 PTS.)
1) A circle has radius 4 feet. A central angle of the circle inter
MIDTERM 2 SOLUTIONS
(CHAPTERS 2 AND 3: POLYNOMIAL, RATIONAL, EXPL, LOG FUNCTIONS)
MATH 141 FALL 2015 KUNIYUKI
150 POINTS TOTAL: 42 FOR PART 1, AND 108 FOR PART 2
PART 1: USING SCIENTIFIC CALCULATORS (42 PTS.)
()
1) Consider g t = 2t 3 8t 2 + 9t 2 . Hint:
QUIZ 1B - SOLUTIONS
(CHAPTER 1: FUNCTIONS)
MATH 141 FALL 2015 KUNIYUKI
60 POINTS TOTAL
No notes or books allowed. A scientific calculator is allowed.
SHORTER PROBLEMS: 25 POINTS TOTAL
()
1) (6 points). Write the domain of f, where f x =
form (the form usi
QUIZ 1A - SOLUTIONS
(CHAPTER 0: PRELIMINARY TOPICS)
MATH 141 FALL 2015 KUNIYUKI
90 POINTS TOTAL
SHORTER PROBLEMS
(29 POINTS: 2 POINTS EACH, UNLESS OTHERWISE SPECIFIED)
1) The symbol means which of the following? (Box in one.)
For all
There exists
Is a mem