FINITE Rings which are not isomorphic to their opposite The 2019 Stack Overflow Developer Survey Results Are In Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)An example of a division ring $D$ that is **not** isomorphic to its opposite ringAn example of a division ring $D$ that is **not** isomorphic to its opposite ringRings with isomorphic proper subringsTwo ordered rings are isomorphic iff their positive semirings are isomorphicRings that are isomorphic to the endomorphism ring of their additive group.What is an example of two k-algebras that are isomorphic as rings, but not as k-algebras?Example of finite ring which is not a Bézout ringNon Commutative rings which are not embeddable in matrix ringsExample of two non-isomorphic, non-trivial rings with the same underlying groupExamples of Finite Non-Unital Integral RingsDedekind domain in which quotient rings are not all finite?
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FINITE Rings which are not isomorphic to their opposite
The 2019 Stack Overflow Developer Survey Results Are In
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)An example of a division ring $D$ that is **not** isomorphic to its opposite ringAn example of a division ring $D$ that is **not** isomorphic to its opposite ringRings with isomorphic proper subringsTwo ordered rings are isomorphic iff their positive semirings are isomorphicRings that are isomorphic to the endomorphism ring of their additive group.What is an example of two k-algebras that are isomorphic as rings, but not as k-algebras?Example of finite ring which is not a Bézout ringNon Commutative rings which are not embeddable in matrix ringsExample of two non-isomorphic, non-trivial rings with the same underlying groupExamples of Finite Non-Unital Integral RingsDedekind domain in which quotient rings are not all finite?
$begingroup$
Please introduce some different (non isomorphic) classes of finite rings which are not isomorphic to their opposite ring. I would like to study some examples.
abstract-algebra ring-theory
asked Apr 8 at 7:11
Sara.TSara.T
19510
$endgroup$
add a comment |
$begingroup$
Please introduce some different (non isomorphic) classes of finite rings which are not isomorphic to their opposite ring. I would like to study some examples.
abstract-algebra ring-theory
asked Apr 8 at 7:11
Sara.TSara.T
19510
$endgroup$
$begingroup$
Are you want big list about it ? If not, see (math.stackexchange.com/questions/45085/…) for a particular example!
$endgroup$
– Chinnapparaj R
Apr 8 at 7:20
$begingroup$
The above link also contains a "biger list", namely this MO-post. James answers there:"Here's a factory for making examples."
$endgroup$
– Dietrich Burde
Apr 8 at 8:08
1
$begingroup$
@ChinnapparajR That example doesn't seem to help: a finite division ring is commutative, hence isomorphic to its opposite ring. But the link contained inside is helpful.
$endgroup$
– rschwieb
Apr 8 at 13:28
add a comment |
$begingroup$
Please introduce some different (non isomorphic) classes of finite rings which are not isomorphic to their opposite ring. I would like to study some examples.
abstract-algebra ring-theory
asked Apr 8 at 7:11
Sara.TSara.T
19510
$endgroup$
Please introduce some different (non isomorphic) classes of finite rings which are not isomorphic to their opposite ring. I would like to study some examples.
abstract-algebra ring-theory
abstract-algebra ring-theory
asked Apr 8 at 7:11
Sara.TSara.T
19510
asked Apr 8 at 7:11
Sara.TSara.T
19510
asked Apr 8 at 7:11
Sara.TSara.T
19510
asked Apr 8 at 7:11
Sara.TSara.T
19510
asked Apr 8 at 7:11
Sara.TSara.T
19510
19510
$begingroup$
Are you want big list about it ? If not, see (math.stackexchange.com/questions/45085/…) for a particular example!
$endgroup$
– Chinnapparaj R
Apr 8 at 7:20
$begingroup$
The above link also contains a "biger list", namely this MO-post. James answers there:"Here's a factory for making examples."
$endgroup$
– Dietrich Burde
Apr 8 at 8:08
1
$begingroup$
@ChinnapparajR That example doesn't seem to help: a finite division ring is commutative, hence isomorphic to its opposite ring. But the link contained inside is helpful.
$endgroup$
– rschwieb
Apr 8 at 13:28
add a comment |
$begingroup$
Are you want big list about it ? If not, see (math.stackexchange.com/questions/45085/…) for a particular example!
$endgroup$
– Chinnapparaj R
Apr 8 at 7:20
$begingroup$
The above link also contains a "biger list", namely this MO-post. James answers there:"Here's a factory for making examples."
$endgroup$
– Dietrich Burde
Apr 8 at 8:08
1
$begingroup$
@ChinnapparajR That example doesn't seem to help: a finite division ring is commutative, hence isomorphic to its opposite ring. But the link contained inside is helpful.
$endgroup$
– rschwieb
Apr 8 at 13:28
$begingroup$
Are you want big list about it ? If not, see (math.stackexchange.com/questions/45085/…) for a particular example!
$endgroup$
– Chinnapparaj R
Apr 8 at 7:20
$begingroup$
Are you want big list about it ? If not, see (math.stackexchange.com/questions/45085/…) for a particular example!
$endgroup$
– Chinnapparaj R
Apr 8 at 7:20
$begingroup$
The above link also contains a "biger list", namely this MO-post. James answers there:"Here's a factory for making examples."
$endgroup$
– Dietrich Burde
Apr 8 at 8:08
$begingroup$
The above link also contains a "biger list", namely this MO-post. James answers there:"Here's a factory for making examples."
$endgroup$
– Dietrich Burde
Apr 8 at 8:08
1
1
$begingroup$
@ChinnapparajR That example doesn't seem to help: a finite division ring is commutative, hence isomorphic to its opposite ring. But the link contained inside is helpful.
$endgroup$
– rschwieb
Apr 8 at 13:28
$begingroup$
@ChinnapparajR That example doesn't seem to help: a finite division ring is commutative, hence isomorphic to its opposite ring. But the link contained inside is helpful.
$endgroup$
– rschwieb
Apr 8 at 13:28
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The only one I have handy is a finite ring that is right Kasch but not left Kasch (so obviously it cannot be isomorphic to its opposite ring, which is left Kasch and not right Kasch.)
You can use any finite field $F$ and then look at the subring of matrices of $M_4(F)$ of the form
beginbmatrix a&0&b&c\ 0&a&0&d\ 0&0&a&0\ 0&0&0&eendbmatrix
The DaRT entry is here, and the original source is given as T.-Y. Lam. Lectures on modules and rings. (2012) @ p 281.
answered Apr 8 at 13:35
rschwiebrschwieb
108k12104253
$endgroup$
1
$begingroup$
Adding this link in case I'm not the only reader who had never heard of Kasch rings.
$endgroup$
– Jyrki Lahtonen
Apr 8 at 13:46
add a comment |
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1 Answer
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oldest
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1 Answer
1
active
oldest
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$begingroup$
The only one I have handy is a finite ring that is right Kasch but not left Kasch (so obviously it cannot be isomorphic to its opposite ring, which is left Kasch and not right Kasch.)
You can use any finite field $F$ and then look at the subring of matrices of $M_4(F)$ of the form
beginbmatrix a&0&b&c\ 0&a&0&d\ 0&0&a&0\ 0&0&0&eendbmatrix
The DaRT entry is here, and the original source is given as T.-Y. Lam. Lectures on modules and rings. (2012) @ p 281.
answered Apr 8 at 13:35
rschwiebrschwieb
108k12104253
$endgroup$
1
$begingroup$
Adding this link in case I'm not the only reader who had never heard of Kasch rings.
$endgroup$
– Jyrki Lahtonen
Apr 8 at 13:46
add a comment |
$begingroup$
The only one I have handy is a finite ring that is right Kasch but not left Kasch (so obviously it cannot be isomorphic to its opposite ring, which is left Kasch and not right Kasch.)
You can use any finite field $F$ and then look at the subring of matrices of $M_4(F)$ of the form
beginbmatrix a&0&b&c\ 0&a&0&d\ 0&0&a&0\ 0&0&0&eendbmatrix
The DaRT entry is here, and the original source is given as T.-Y. Lam. Lectures on modules and rings. (2012) @ p 281.
answered Apr 8 at 13:35
rschwiebrschwieb
108k12104253
$endgroup$
1
$begingroup$
Adding this link in case I'm not the only reader who had never heard of Kasch rings.
$endgroup$
– Jyrki Lahtonen
Apr 8 at 13:46
add a comment |
$begingroup$
The only one I have handy is a finite ring that is right Kasch but not left Kasch (so obviously it cannot be isomorphic to its opposite ring, which is left Kasch and not right Kasch.)
You can use any finite field $F$ and then look at the subring of matrices of $M_4(F)$ of the form
beginbmatrix a&0&b&c\ 0&a&0&d\ 0&0&a&0\ 0&0&0&eendbmatrix
The DaRT entry is here, and the original source is given as T.-Y. Lam. Lectures on modules and rings. (2012) @ p 281.
answered Apr 8 at 13:35
rschwiebrschwieb
108k12104253
$endgroup$
The only one I have handy is a finite ring that is right Kasch but not left Kasch (so obviously it cannot be isomorphic to its opposite ring, which is left Kasch and not right Kasch.)
You can use any finite field $F$ and then look at the subring of matrices of $M_4(F)$ of the form
beginbmatrix a&0&b&c\ 0&a&0&d\ 0&0&a&0\ 0&0&0&eendbmatrix
The DaRT entry is here, and the original source is given as T.-Y. Lam. Lectures on modules and rings. (2012) @ p 281.
answered Apr 8 at 13:35
rschwiebrschwieb
108k12104253
answered Apr 8 at 13:35
rschwiebrschwieb
108k12104253
answered Apr 8 at 13:35
rschwiebrschwieb
108k12104253
answered Apr 8 at 13:35
rschwiebrschwieb
108k12104253
108k12104253
1
$begingroup$
Adding this link in case I'm not the only reader who had never heard of Kasch rings.
$endgroup$
– Jyrki Lahtonen
Apr 8 at 13:46
add a comment |
1
$begingroup$
Adding this link in case I'm not the only reader who had never heard of Kasch rings.
$endgroup$
– Jyrki Lahtonen
Apr 8 at 13:46
1
1
$begingroup$
Adding this link in case I'm not the only reader who had never heard of Kasch rings.
$endgroup$
– Jyrki Lahtonen
Apr 8 at 13:46
$begingroup$
Adding this link in case I'm not the only reader who had never heard of Kasch rings.
$endgroup$
– Jyrki Lahtonen
Apr 8 at 13:46
add a comment |
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$begingroup$
Are you want big list about it ? If not, see (math.stackexchange.com/questions/45085/…) for a particular example!
$endgroup$
– Chinnapparaj R
Apr 8 at 7:20
$begingroup$
The above link also contains a "biger list", namely this MO-post. James answers there:"Here's a factory for making examples."
$endgroup$
– Dietrich Burde
Apr 8 at 8:08
1
$begingroup$
@ChinnapparajR That example doesn't seem to help: a finite division ring is commutative, hence isomorphic to its opposite ring. But the link contained inside is helpful.
$endgroup$
– rschwieb
Apr 8 at 13:28