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Order in the Integers

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?.

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Order in the Integers

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  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 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?

  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 => a = 0 or b = 0 for all a, b  R?

  6. 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)?

  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 => 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 A={A,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 => 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 => 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 M3={0,1,2}. Define module 3 + and • in the usual way, which is indicated in the following Cayley tables.

  13. 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?

  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 > b if a is to right of b on number line • a > b if a – b  Z+   -3 -2 -1 0 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 > 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.

  25. 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

  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 =2m+n and2m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?

  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? • Conjecture:Unity 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+ = {m•1: m  Z} and D+= { me: m  Z}

  35. Thank You !!!!

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