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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 Ru,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

w u - 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

w u - 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 gt a 0 or b 0 for all

a, b ? R?

Power Set

- ?(A ) is the set of all subsets of A with

ab(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 gt 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 non-zero

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 gt 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 AA?,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 gt 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 gt 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 M30,1,2. Define module 3 and in the

usual way, which is indicated in the following

Cayley tables.

M30,1,2 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, IndiaDied 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 gt b if a is to right of b on number line
- a gt b if a b ? Z

?

?

0

1

2

3

-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 gt d if c - d ? D.
- Clearly by this definition
- a gt 0 gt a ? D
- a lt 0 gt -a ? D
- We can now prove most simple inequality

properties.

Examples

- a gt b gt a c gt b c, ? c ? D
- a gt b and c gt 0 gt ac gt bc
- a gt b and c lt 0 gt ac lt bc
- a gt b and b gt c gt a gt 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 2mn and 2m?2n 2mn
- 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? - ConjectureUnity 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 m1 m ? Z and
- D m?e m ? Z

- Thank You !!!!