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## Order in the Integers

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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. – PowerPoint PPT presentation

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Title: Order in the Integers

1
Order in the Integers
• Characterization of the
• Ring of Integers

2
• 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?

3
Exploration
• Let (R,,?) be a commutative ring with unity.
Let c,d ? R where c?? 0 and d ? 0.
• Can c?d 0?

4
• 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?

5
• 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?

6
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)?

7
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?

8
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?

9
• 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?

10
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

11
Integral Domain
• A ring D with more than one element that has
• Commutative
• Unity
• No non-zero divisors of zero
• r s 0 gt r 0 or s 0.

12
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.

13
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?

14
Brahmagupta
• Born 598 in (possibly) Ujjain, IndiaDied 670
in India

15
• 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.

16
• He also gives arithmetical rules in terms of
fortunes (positive numbers) and debts (negative
numbers)-

17
• 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.

18
• 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.

19
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
?
?
?
?
?
?
20
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
• 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).

21
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?

22
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?

23
Exploration
• Are the even integers an ordered integral domain?
Are they an ordered ring?

24
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.

25
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

26
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?

27
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.

28
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?

29
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?

30
• How do we show any (D,?,?) is isomorphic to the
integers (Z,,)?

31
• How can we formulate a general expression for all
the elements a ? D so we can determine a map?

32
• 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.

33
• How can we use e to characterize other elements
of D ?

34
• So how can we define our mapping
• f Z ? D where Z m1 m ? Z and
• D m?e m ? Z

35
• Thank You !!!!