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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… Hermione Granger quoting
Wilbert Slinkhard
Conjecture: All odd numbers greater than 1 are prime. when they want to make them sound more interesting.
Counterexample: The number15
is not prime
Conjecture: when they want to make them sound more interesting. All odd numbers greater than 1 are prime.
Counterexample: The number 15
is not prime
Conjecture: All odd numbers greater than 1 are prime. when they want to make them sound more interesting.
Counterexample: The number 15
is not prime
Conjecture: All odd numbers greater than 1 are prime. when they want to make them sound more interesting.
Counterexample: The number 15
is an odd number greater than 1 and it is not prime.
Definition: when they want to make them sound more interesting. 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: when they want to make them sound more interesting.
A ring R is a set with a commutative addition, a subtraction, a multiplication and which contains the elements 0 and (sometimes) 1.
Examples: when they want to make them sound more interesting.
Examples: when they want to make them sound more interesting.
Nonexamples: when they want to make them sound more interesting.
Nonexamples: when they want to make them sound more interesting.
A Counterexample when they want to make them sound more interesting. in Ring Theory is a ring which is a counterexample to a conjecture about rings.
Examples: when they want to make them sound more interesting.
Conjecture: when they want to make them sound more interesting. The multiplication in a ring is commutative.
Conjecture: when they want to make them sound more interesting. The multiplication in a ring is commutative.
Counterexample: M2()
Conjecture: when they want to make them sound more interesting. The multiplication in a ring is commutative.
Counterexample: M2()
Why?
Conjecture: when they want to make them sound more interesting. The multiplication in a ring is commutative.
Counterexample: M2()
Why?
Conclusion: when they want to make them sound more interesting. Not every ring is commutative.
Examples: when they want to make them sound more interesting.
Conjecture: when they want to make them sound more interesting. For any nonzero elements a and b in R, there is a unique element c of R so that a=bc.
Conjecture: when they want to make them sound more interesting. 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: when they want to make them sound more interesting. 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: when they want to make them sound more interesting. 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: when they want to make them sound more interesting. 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: when they want to make them sound more interesting. 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: when they want to make them sound more interesting. 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: when they want to make them sound more interesting. Every ring can be embedded in a division ring.
Conjecture: when they want to make them sound more interesting. Every ring can be embedded in a division ring.
Counterexample: M2()
Conjecture: when they want to make them sound more interesting. Every ring can be embedded in a division ring.
Counterexample: M2()
Why?
Conjecture: when they want to make them sound more interesting. Every ring can be embedded in a division ring.
Counterexample: M2()
Why?
Conjecture: when they want to make them sound more interesting. Every ring can be embedded in a division ring.
Counterexample: M2()
Why?
Definition: when they want to make them sound more interesting. A ring R is von Neumann Regular if for every xR there is an element yR so that xyx=x.
Definition: when they want to make them sound more interesting. 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: when they want to make them sound more interesting. 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: when they want to make them sound more interesting. 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 when they want to make them sound more interesting. : Every ring is von Neumann regular.
Conjecture when they want to make them sound more interesting. : Every ring is von Neumann regular.
Counterexample:
Conjecture when they want to make them sound more interesting. : Every ring is von Neumann regular.
Counterexample:
Why?
Conjecture when they want to make them sound more interesting. : 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: when they want to make them sound more interesting. Every von Neumann regular ring is a division ring.
Conjecture: when they want to make them sound more interesting. Every von Neumann regular ring is a division ring.
Counterexample 1:M2() is von Neumann regular.
Definition when they want to make them sound more interesting. : 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 when they want to make them sound more interesting. : 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: when they want to make them sound more interesting. Every von Neumann regular ring is a division ring.
Counterexample 2:
Conjecture: when they want to make them sound more interesting. Every von Neumann regular ring is a division ring.
Counterexample 2:
Why?
Conjecture: when they want to make them sound more interesting. Every von Neumann regular ring is a division ring.
Counterexample 2:
Why?
is von Neumann Regular:
Suppose (a,b) .
Conjecture: when they want to make them sound more interesting. 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: when they want to make them sound more interesting. 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 when they want to make them sound more interesting.
Isomorphic Rings when they want to make them sound more interesting.
Question: when they want to make them sound more interesting. Is ?
Question: when they want to make them sound more interesting. Is ?
Answer: No
Why? i2=-1, but (0,1)(0,1)=(0,1)
Question: when they want to make them sound more interesting. 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 when they want to make them sound more interesting.
Conjecture: when they want to make them sound more interesting. For rings A, B, and C
Conjecture: when they want to make them sound more interesting. For rings A, B, and C
Counterexample:
Conjecture: when they want to make them sound more interesting. For rings A, B, and C
Counterexample:
Why?
Conjecture: when they want to make them sound more interesting. For rings A, B, and C
Counterexample:
Why?R0
Conjecture: when they want to make them sound more interesting. If A and B are von Neumann regular rings, then
Conjecture: when they want to make them sound more interesting. If A and B are von Neumann regular rings, then
Counterexample: Unknown.
Conjecture: when they want to make them sound more interesting. If A and B are von Neumann regular rings, then
Counterexample: Unknown.
This is an open problem!
Published: 1974 when they want to make them sound more interesting.
Rank: 31,815
Published: 1974 when they want to make them sound more interesting.
Rank: 31,815
Published: 1978
Rank: 190,752
Question: when they want to make them sound more interesting. Which examples did not have a 1?
Examples: when they want to make them sound more interesting.
Question: when they want to make them sound more interesting. Which examples did not have a 1?
Answer:
Question: when they want to make them sound more interesting. Which examples did not have a 1?
Answer:
Question: when they want to make them sound more interesting. Which examples did not have a 1?
Answer: