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Counterexamples in Ring Theory Kathi Crow Gettysburg College Connecticut College November 7, 2005

Counterexamples in Ring Theory Kathi Crow Gettysburg College Connecticut College November 7, 2005. “Counter-jinx is just a name people give their jinxes when they want to make them sound more acceptable.” .

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Counterexamples in Ring Theory Kathi Crow Gettysburg College Connecticut College November 7, 2005

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  1. Counterexamples in Ring TheoryKathi CrowGettysburg CollegeConnecticut CollegeNovember 7, 2005

  2. “Counter-jinx is just a name people give their jinxes when they want to make them sound more acceptable.”

  3. “Counter-jinx is just a name people give their jinxes when they want to make them sound more acceptable.” … Hermione Granger quoting Wilbert Slinkhard

  4. Counterexample is just a name people give their examples when they want to make them sound more interesting.

  5. Conjecture: All odd numbers greater than 1 are prime. Counterexample: The number15 is not prime

  6. Conjecture: All odd numbers greater than 1 are prime. Counterexample: The number 15 is not prime

  7. Conjecture: All odd numbers greater than 1 are prime. Counterexample: The number 15 is not prime

  8. Conjecture: All odd numbers greater than 1 are prime. Counterexample: The number 15 is an odd number greater than 1 and it is not prime.

  9. Definition: The set R together with operations + and • is called a Ring if the following properties hold for every selection of elements a,b,c in R: • a+b=b+a • (a+b)+c=a+(b+c) • There is an element 0 in R so that a+0=a • For any element a of R, there is an element x of R so that x+a=0 • (a•b)•c=a•(b•c) • a•(b+c)=a•b+a•c and (b+c)•a=b•a+c•a • (There is an element 1 in R so that a•1=a=1•a)

  10. Informal Definition: A ring R is a set with a commutative addition, a subtraction, a multiplication and which contains the elements 0 and (sometimes) 1.

  11. Examples:

  12. Examples: • The real numbers  • The complex numbers  • The rational numbers  • The integers  • Even integers • Matrices Mn()

  13. Nonexamples:

  14. Nonexamples: • Natural Numbers  • Odd Integers • Vector Spaces • Polynomials of degree 2 or less

  15. A Counterexample in Ring Theory is a ring which is a counterexample to a conjecture about rings.

  16. Examples: • The real numbers  • The complex numbers  • The rational numbers  • The integers  • Even integers • Matrices Mn()

  17. Conjecture: The multiplication in a ring is commutative.

  18. Conjecture: The multiplication in a ring is commutative. Counterexample: M2()

  19. Conjecture: The multiplication in a ring is commutative. Counterexample: M2() Why?

  20. Conjecture: The multiplication in a ring is commutative. Counterexample: M2() Why?

  21. Conclusion: Not every ring is commutative.

  22. Examples: • The real numbers  • The complex numbers  • The rational numbers  • The integers  • Even integers • Matrices Mn()

  23. Conjecture: For any nonzero elements a and b in R, there is a unique element c of R so that a=bc.

  24. Conjecture: For any nonzero elements a and b in R, there is a unique element c of R so that a=bc. (In other words, we can divide in R.)

  25. Conjecture: For any nonzero elements a and b in R, there is a unique element c of R so that a=bc. (In other words, we can divide in R.) Counterexample: 

  26. Conjecture: For any nonzero elements a and b in R, there is a unique element c of R so that a=bc. (In other words, we can divide in R.) Counterexample:  Why?

  27. Conjecture: For any nonzero elements a and b in R, there is a unique element c of R so that a=bc. (In other words, we can divide in R.) Counterexample:  Why? Because 23 is not an integer.

  28. Conjecture: For any nonzero elements a and b in R, there is a unique element c of R so that a=bc. (In other words, we can divide in R.) Counterexample:  Why? Because 23 is not an integer. (There is no integer c so that 2=3c.)

  29. Conjecture: For any nonzero elements a and b in R, there is a unique element c of R so that a=bc. (In other words, we can divide in R.) Counterexample:  Why? Because 23 is not an integer. (There is no integer c so that 2=3c.) The ring  is not a division ring.

  30. Conjecture: Every ring can be embedded in a division ring.

  31. Conjecture: Every ring can be embedded in a division ring. Counterexample: M2()

  32. Conjecture: Every ring can be embedded in a division ring. Counterexample: M2() Why?

  33. Conjecture: Every ring can be embedded in a division ring. Counterexample: M2() Why?

  34. Conjecture: Every ring can be embedded in a division ring. Counterexample: M2() Why?

  35. Conclusions: • Not every ring is commutative. • Not every ring can be embedded in a division ring.

  36. Definition: A ring R is von Neumann Regular if for every xR there is an element yR so that xyx=x.

  37. Definition: A ring R is von Neumann Regular if for every xR there is an element yR so that xyx=x. We say y is a quasi-inverse for x.

  38. Definition: A ring R is von Neumann Regular if for every xR there is an element yR so that xyx=x. We say y is a quasi-inverse for x.

  39. Definition: A ring R is von Neumann Regular if for every xR there is an element yR so that xyx=x. We say y is a quasi-inverse for x. Note: Every division ring is von Neumann Regular. (Let y=x-1 or y=0.)

  40. Conjecture: Every ring is von Neumann regular.

  41. Conjecture: Every ring is von Neumann regular. Counterexample: 

  42. Conjecture: Every ring is von Neumann regular. Counterexample:  Why?

  43. Conjecture: Every ring is von Neumann regular. Counterexample:  Why? Suppose there is a y so that 2y2=2. The only real number for which this equation holds is y=½. Since ½ is not an integer, 2 does not have a quasi-inverse in .

  44. Conjecture: Every von Neumann regular ring is a division ring.

  45. Conjecture: Every von Neumann regular ring is a division ring. Counterexample 1:M2() is von Neumann regular.

  46. Definition: If A and B are rings then the direct sum of A and B is the ring AB= {(a,b): aA and bB} with addition and multiplication defined coordinatewise.

  47. Definition: If A and B are rings then the direct sum of A and B is the ring AB= {(a,b): aA and bB} with addition and multiplication defined coordinatewise. Example: Consider . (π,2)+(1,-3)=(π+1,2-3)=(π+1,-1) (π,2)(1,-3)=(π•1,2(-3)) =(π,-6)

  48. Conjecture: Every von Neumann regular ring is a division ring. Counterexample 2:

  49. Conjecture: Every von Neumann regular ring is a division ring. Counterexample 2: Why?

  50. Conjecture: Every von Neumann regular ring is a division ring. Counterexample 2: Why?  is von Neumann Regular: Suppose (a,b) .

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