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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 l.jpg

Counterexamples in Ring TheoryKathi CrowGettysburg CollegeConnecticut CollegeNovember 7, 2005


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


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

… Hermione Granger quoting

Wilbert Slinkhard


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


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Conjecture: All odd numbers greater than 1 are prime.

Counterexample: The number15

is not prime


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Conjecture: All odd numbers greater than 1 are prime.

Counterexample: The number 15

is not prime


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Conjecture: All odd numbers greater than 1 are prime.

Counterexample: The number 15

is not prime


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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.


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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)


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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.


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Examples:


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Examples:

  • The real numbers 

  • The complex numbers 

  • The rational numbers 

  • The integers 

  • Even integers

  • Matrices Mn()


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Nonexamples:


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Nonexamples:

  • Natural Numbers 

  • Odd Integers

  • Vector Spaces

    • Polynomials of degree 2 or less


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


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Examples:

  • The real numbers 

  • The complex numbers 

  • The rational numbers 

  • The integers 

  • Even integers

  • Matrices Mn()


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Conjecture: The multiplication in a ring is commutative.


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Conjecture: The multiplication in a ring is commutative.

Counterexample: M2()


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Conjecture: The multiplication in a ring is commutative.

Counterexample: M2()

Why?


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Conjecture: The multiplication in a ring is commutative.

Counterexample: M2()

Why?


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Conclusion: Not every ring is commutative.


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Examples:

  • The real numbers 

  • The complex numbers 

  • The rational numbers 

  • The integers 

  • Even integers

  • Matrices Mn()


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


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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.)


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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: 


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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?


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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.


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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.)


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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.


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


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

Counterexample: M2()


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

Counterexample: M2()

Why?


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

Counterexample: M2()

Why?


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

Counterexample: M2()

Why?


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Conclusions:

  • Not every ring is commutative.

  • Not every ring can be embedded in a division ring.


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


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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.


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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.


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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.)


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Conjecture: Every ring is von Neumann regular.


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Conjecture: Every ring is von Neumann regular.

Counterexample: 


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Conjecture: Every ring is von Neumann regular.

Counterexample: 

Why?


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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 .


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


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

Counterexample 1:M2() is von Neumann regular.


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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.


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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)


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

Counterexample 2:


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

Counterexample 2:

Why?


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

Counterexample 2:

Why?

 is von Neumann Regular:

Suppose (a,b) .


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

Counterexample 2:

Why?

 is von Neumann Regular:

Suppose (a,b) . If a=0, then let c=0; otherwise let c=a-1. Similarly d=0 or d=b-1. Now (a,b)(c,d)(a,b)=(a,b).


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

Counterexample 2:

Why?

 is von Neumann Regular:

Suppose (a,b) . If a=0, then let c=0; otherwise let c=a-1. Similarly d=0 or d=b-1. Now (a,b)(c,d)(a,b)=(a,b).

 is not a division ring:

Since (1,0)(0,1)=(0,0) and (1,0)(0,4)=(0,0) there is no good way to define (0,0)(1,0).


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Conclusions:

  • Not every ring is commutative.

  • Not every ring can be embedded in a division ring.

  • Every division ring is von Neumann regular

  • Not every von Neumann regular ring is a division ring.


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Isomorphic Rings


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Isomorphic Rings


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Question: Is ?


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Question: Is ?

Answer: No

Why?


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Question: Is ?

Answer: No

Why? i2=-1, but (0,1)(0,1)=(0,1)


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Question: Is ?

Answer: No

Why? i2=-1, but (0,1)(0,1)=(0,1)

Better Answer: is a division ring.

 is not a division ring since

(1,0)(0,1)=(0,0)


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Cancellation Properties


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Conjecture: For rings A, B, and C


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Conjecture: For rings A, B, and C

Counterexample:


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Conjecture: For rings A, B, and C

Counterexample:

Why?


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Conjecture: For rings A, B, and C

Counterexample:

Why?R0


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Conjecture: If A and B are von Neumann regular rings, then


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Conjecture: If A and B are von Neumann regular rings, then

Counterexample: Unknown.


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Conjecture: If A and B are von Neumann regular rings, then

Counterexample: Unknown.

This is an open problem!


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Published: 1974

Rank: 31,815


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Published: 1974

Rank: 31,815

Published: 1978

Rank: 190,752


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Question:Which examples did not have a 1?


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Examples:

  • The real numbers 

  • The complex numbers 

  • The rational numbers 

  • The integers 

  • Even integers

  • Matrices Mn()

  • 

  • ···


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Question:Which examples did not have a 1?

Answer:


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Question:Which examples did not have a 1?

Answer:

  • Even Integers


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Question:Which examples did not have a 1?

Answer:

  • Even Integers

  • ···


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