Counterexamples in ring theory kathi crow gettysburg college connecticut college november 7 2005
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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 TheoryKathi CrowGettysburg CollegeConnecticut CollegeNovember 7, 2005


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


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

… Hermione Granger quoting

Wilbert Slinkhard


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


Conjecture: All odd numbers greater than 1 are prime.

Counterexample: The number15

is not prime


Conjecture: All odd numbers greater than 1 are prime.

Counterexample: The number 15

is not prime


Conjecture: All odd numbers greater than 1 are prime.

Counterexample: The number 15

is not prime


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.


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)


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.


Examples:


Examples:

  • The real numbers 

  • The complex numbers 

  • The rational numbers 

  • The integers 

  • Even integers

  • Matrices Mn()


Nonexamples:


Nonexamples:

  • Natural Numbers 

  • Odd Integers

  • Vector Spaces

    • Polynomials of degree 2 or less


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


Examples:

  • The real numbers 

  • The complex numbers 

  • The rational numbers 

  • The integers 

  • Even integers

  • Matrices Mn()


Conjecture: The multiplication in a ring is commutative.


Conjecture: The multiplication in a ring is commutative.

Counterexample: M2()


Conjecture: The multiplication in a ring is commutative.

Counterexample: M2()

Why?


Conjecture: The multiplication in a ring is commutative.

Counterexample: M2()

Why?


Conclusion: Not every ring is commutative.


Examples:

  • The real numbers 

  • The complex numbers 

  • The rational numbers 

  • The integers 

  • Even integers

  • Matrices Mn()


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


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.)


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: 


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?


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.


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.)


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.


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


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

Counterexample: M2()


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

Counterexample: M2()

Why?


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

Counterexample: M2()

Why?


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

Counterexample: M2()

Why?


Conclusions:

  • Not every ring is commutative.

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


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


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.


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.


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.)


Conjecture: Every ring is von Neumann regular.


Conjecture: Every ring is von Neumann regular.

Counterexample: 


Conjecture: Every ring is von Neumann regular.

Counterexample: 

Why?


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 .


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


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

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


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.


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)


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

Counterexample 2:


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

Counterexample 2:

Why?


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

Counterexample 2:

Why?

 is von Neumann Regular:

Suppose (a,b) .


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).


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).


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.


Isomorphic Rings


Isomorphic Rings


Question: Is ?


Question: Is ?

Answer: No

Why?


Question: Is ?

Answer: No

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


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)


Cancellation Properties


Conjecture: For rings A, B, and C


Conjecture: For rings A, B, and C

Counterexample:


Conjecture: For rings A, B, and C

Counterexample:

Why?


Conjecture: For rings A, B, and C

Counterexample:

Why?R0


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


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

Counterexample: Unknown.


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

Counterexample: Unknown.

This is an open problem!


Published: 1974

Rank: 31,815


Published: 1974

Rank: 31,815

Published: 1978

Rank: 190,752


Question:Which examples did not have a 1?


Examples:

  • The real numbers 

  • The complex numbers 

  • The rational numbers 

  • The integers 

  • Even integers

  • Matrices Mn()

  • 

  • ···


Question:Which examples did not have a 1?

Answer:


Question:Which examples did not have a 1?

Answer:

  • Even Integers


Question:Which examples did not have a 1?

Answer:

  • Even Integers

  • ···


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