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 cd = 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 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 xWhat 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=(ab)\(ab) 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 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 => 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 a0 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.

+ 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?

M3={0,1,2} Cayley tables for operations

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+

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 > 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 2m2n = 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+= { me: m Z}

Thank You !!!!