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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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“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:
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:
Nonexamples:
Nonexamples:
A Counterexample in Ring Theory is a ring which is a counterexample to a conjecture about rings.
Examples:
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:
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 23 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 23 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 23 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?
Definition: A ring R is von Neumann Regular if for every xR there is an element yR so that xyx=x.
Definition: A ring R is von Neumann Regular if for every xR there is an element yR so that xyx=x. We say y is a quasi-inverse for x.
Definition: A ring R is von Neumann Regular if for every xR there is an element yR so that xyx=x. We say y is a quasi-inverse for x.
Definition: A ring R is von Neumann Regular if for every xR there is an element yR 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
AB= {(a,b): aA and bB}
with addition and multiplication defined coordinatewise.
Definition: If A and B are rings then the direct sum of A and B is the ring
AB= {(a,b): aA and bB}
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).
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?R0
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:
Question:Which examples did not have a 1?
Answer:
Question:Which examples did not have a 1?
Answer:
Question:Which examples did not have a 1?
Answer: