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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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Date added: 15 May 2020
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Learn more at: http://www.math.wvu.edu
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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
    three additional properties
  • 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
    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).

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 !!!!
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