Download Lecture4 - WVU Math Department

Order in the Integers
Characterization of the
Ring of Integers
Let Z be the set of integers and +,  be
the binary operations of integer addition
and multiplication.
(Z,+,) is a commutative ring with unity
What other properties of (Z,+,) distinguish
it from other rings?
Exploration
• Let (R,+,) be a commutative ring with
unity. Let c,d  R where c 0 and d  0.
• Can cd = 0?
Let R={u,v,w,x}
Define addition and multiplication by the
Cayley tables:
+ u v w x
• u v w x
u u v w x
u u u u u
v v u x w
v u v w x
w w x u v
w u w wu
x x w v u
x u x u x
Is (R,+, • ) a commutative ring with unity?
+ u v w x
• u v w x
u u v w x
u u u u u
v v u x w
v u v w x
w w x u v
w u w wu
x x w v u
x u x u x
What is the additive identity?
What is the unity (multiplicative identity)?
Does a • b = 0 => a = 0 or b = 0 for all
a, b  R?
Power Set
(A ) is the set of all subsets of A with
a+b=(ab)\(ab) and a • b = a  b.
• Recall what the zero and unity are for the
power set ring.
• Does a • b = 0 => a = 0 or b = 0 for all
a, b  (A)?
Divisor Of Zero
a  R is a divisor of zero in R if  b  R 
a • b = 0 or b • a = 0?
• Is the zero of R a divisor of zero?
• Does the ring of integers have any nonzero divisors of zero?
Cancellation Law Of
Multiplication
We often use the Cancellation Law to
solve equations.
If a,b,c  ring R, then ab = ac => b = c
• What restriction must be placed on a for
this statement to hold?
• Suppose a is a non-zero divisor of zero,
does this law hold?
Example: Let A={A,K ,Q ,J }.
Consider ((A), + , • ).
Given
{A,K} • {K,Q} = {A,K} • {K,J }
So a • b = a • c
Does b = c?
Cancellation Law Proof
Prove: If a,b,c  ring R and a0 is not a
divisor of zero, then ab = ac => b = c
Proof:
Integral Domain
A ring D with more than one element that
has three additional properties:
• Commutative
• Unity
• No non-zero divisors of zero:
r • s = 0 => r = 0 or s = 0.
Exploration
• Are the integers the only example of an
integral domain? Consider other number
sets you are familiar with such as the
rational numbers, the real numbers, or the
complex numbers.
• Let M3={0,1,2}. Define module 3 + and •
in the usual way, which is indicated in the
following Cayley tables.
M3={0,1,2} Cayley tables for operations
+ 0 1 2
• 0 1 2
0 0 1 2
0 0 0 0
1 1 2 0
1 0 1 2
2 2 0 1
2 0 2 1
a + b = c mod 3 a • b = d mod 3
• Is (M3,+, •) an integral domain?
• How does (M3,+, •) differ in structure from
the integral domain of integers?
Brahmagupta
Born: 598 in (possibly) Ujjain,
India
Died: 670 in India
• Brahmagupta's understanding of the number
systems went far beyond that of others of the
period. In the Brahmasphutasiddhanta he
defined zero as the result of subtracting a
number from itself. He gave some properties as
follows:
• When zero is added to a number or subtracted
from a number, the number remains unchanged;
and a number multiplied by zero becomes zero.
• He also gives arithmetical rules in terms of
fortunes (positive numbers) and debts
(negative numbers):-











A debt minus zero is a debt.
A fortune minus zero is a fortune.
Zero minus zero is a zero.
A debt subtracted from zero is a fortune.
A fortune subtracted from zero is a debt.
The product of zero multiplied by a debt or fortune is
zero.
The product of zero multiplied by zero is zero.
The product or quotient of two fortunes is one fortune.
The product or quotient of two debts is one fortune.
The product or quotient of a debt and a fortune is a debt.
The product or quotient of a fortune and a debt is a debt.
• Brahmagupta then tried to extend arithmetic
to include division by zero:• Positive or negative numbers when divided
by zero is a fraction the zero as denominator.
Zero divided by negative or positive numbers
is either zero or is expressed as a fraction
with zero as numerator and the finite quantity
as denominator.
• Zero divided by zero is zero.
Order For Integers
• Integers can be arranged in order on a
number line
• a > b if a is to right of b on number line
• a > b if a – b  Z+





-3 -2 -1 0 1 2
3



Ordered Integral Domain
An integral domain D that contains a
subset D+ with three properties.
1. If a, b  D+ then a + b  D+ ( Closure
with respect to Addition).
2. If a, b  D + then a • b  D+ (Closure
with respect to Multiplication).
3.  a  D exactly one of the following
holds: a = 0, a  D+ , -a  D+ (Trichotomy
Law).
Ordered Integral Domain of
Integers
• Verify that (Z,+,•) is an ordered integral
domain.
• Are the Rational Numbers an ordered
integral domain?
• The Real Numbers?
• The Complex Numbers?
Exploration
• Is (M3,+,•) an ordered integral domain?
+ 0 1 2
• 0 1 2
0 0 1 2
0 0 0 0
1 1 2 0
1 0 1 2
2 2 0 1
2 0 2 1
• Can any finite ring ever be an ordered integral
domain?
Exploration
• Are the even integers an ordered integral
domain? Are they an ordered ring?
Order Relation
Let c, d  D. Define c > d if c - d  D+.
Clearly by this definition:
• a > 0 => a  D +
• a < 0 => -a  D +
We can now prove most simple inequality
properties.
Examples
•
•
•
•
a > b => a + c > b + c,  c  D
a > b and c > 0 => ac > bc
a > b and c < 0 => ac < bc
a > b and b > c => a > c
Well-Ordered Set
A set S of elements of an ordered integral
domain is well-ordered if each non-empty
U  S contains a least element a, such
that  x  U, a  x.
• Which set in Z is well -ordered, Z+ or Z - ?
• What is the least element in the well ordered set?
• Are the Rational Numbers well-ordered?
Characterization of the Integers
• The only ordered integral domain in which
the positive set is well-ordered is the ring
of integers.
• Any other ordered integral domain with a
well ordered positive set is isomorphic to
(Z,+,•)
• Well-ordered property is equivalent to the
induction principle - so induction is a
characteristic of the positive integers.
Exploration
Let D = 2n, n  Z.
Define 2m  2n = 2m+n and 2m2n = 2m•n
• Is this an ordered integral domain with a
well-ordered positive set?
• Relate it to the ring of integers – what
does it mean to be isomorphic?
Verification
(Z,+,•) is the only ordered integral domain
in which the set of positive elements is
well ordered up to isomorphism.
• What does up to isomorphism mean?
• How do we show any (D,,) is
isomorphic to the integers (Z,+,•)?
• How can we formulate a general
expression for all the elements a  D+ so
we can determine a map?
• How can we extend this idea to other
(D,,)?
• What is the smallest element of D+ for any
OID with a well-ordered positive subset?
• Conjecture:Unity is smallest element of D+
so it is our building block.
• How can we use e to characterize other
elements of D+ ?
• So how can we define our mapping
f: Z  D where Z+ = {m•1: m  Z} and
D+= { me: m  Z}
Thank You !!!!