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# Counterexamples in Ring Theory Kathi Crow Gettysburg College Connecticut College November 7, 2005 - PowerPoint PPT Presentation

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

… Hermione Granger quoting

## 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 ?

Why?

Question: Is ?

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

Question: Is ?

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?

Question:Which examples did not have a 1?