a and b associates implies that $exists uin A$ unit such that $a=bu$
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I know that in a domain $A$, $a,bin A$ are associates if and only if there exists a unit $uin A$ such that $a=bu$. But, when A is not a domain, but a general unitary and commutative ring, we can only prove the second implication.
However, I haven't been able to find a counterexample ( a pair a,b in a unitary and commutative ring such that a and b are associates but there is no unit u such that $a=bu$). Of course, as I was told the biconditional was only true in domains, I tried looking at rings $mathbb{Z}_n$ with n no prime. But, the double implication seems to hold for all of them.
My question are:
1)Is it true the double implication in every $mathbb{Z}_n$? Does it hold for any principal ideal ring? If the answer is yes, how can I prove it?
2)Can you give a counterexample for the double implication?
abstract-algebra ring-theory
edited Nov 15 at 18:19
mvw
31.2k22252
asked Nov 15 at 18:01
Seven
829
add a comment |
up vote
0
down vote
favorite
I know that in a domain $A$, $a,bin A$ are associates if and only if there exists a unit $uin A$ such that $a=bu$. But, when A is not a domain, but a general unitary and commutative ring, we can only prove the second implication.
However, I haven't been able to find a counterexample ( a pair a,b in a unitary and commutative ring such that a and b are associates but there is no unit u such that $a=bu$). Of course, as I was told the biconditional was only true in domains, I tried looking at rings $mathbb{Z}_n$ with n no prime. But, the double implication seems to hold for all of them.
My question are:
1)Is it true the double implication in every $mathbb{Z}_n$? Does it hold for any principal ideal ring? If the answer is yes, how can I prove it?
2)Can you give a counterexample for the double implication?
abstract-algebra ring-theory
edited Nov 15 at 18:19
mvw
31.2k22252
asked Nov 15 at 18:01
Seven
829
See this question and this one. For $mathbb{Z}_n$, see this question.
– J.-E. Pin
Nov 15 at 18:15
Okay, I see there are counterexamples but they are not trivial. About $mathbb{Z}_n$, I don't see that last topic solve my question
– Seven
Nov 15 at 18:26
I got it thank you
– Seven
Nov 15 at 19:00
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I know that in a domain $A$, $a,bin A$ are associates if and only if there exists a unit $uin A$ such that $a=bu$. But, when A is not a domain, but a general unitary and commutative ring, we can only prove the second implication.
However, I haven't been able to find a counterexample ( a pair a,b in a unitary and commutative ring such that a and b are associates but there is no unit u such that $a=bu$). Of course, as I was told the biconditional was only true in domains, I tried looking at rings $mathbb{Z}_n$ with n no prime. But, the double implication seems to hold for all of them.
My question are:
1)Is it true the double implication in every $mathbb{Z}_n$? Does it hold for any principal ideal ring? If the answer is yes, how can I prove it?
2)Can you give a counterexample for the double implication?
abstract-algebra ring-theory
edited Nov 15 at 18:19
mvw
31.2k22252
asked Nov 15 at 18:01
Seven
829
I know that in a domain $A$, $a,bin A$ are associates if and only if there exists a unit $uin A$ such that $a=bu$. But, when A is not a domain, but a general unitary and commutative ring, we can only prove the second implication.
However, I haven't been able to find a counterexample ( a pair a,b in a unitary and commutative ring such that a and b are associates but there is no unit u such that $a=bu$). Of course, as I was told the biconditional was only true in domains, I tried looking at rings $mathbb{Z}_n$ with n no prime. But, the double implication seems to hold for all of them.
My question are:
1)Is it true the double implication in every $mathbb{Z}_n$? Does it hold for any principal ideal ring? If the answer is yes, how can I prove it?
2)Can you give a counterexample for the double implication?
abstract-algebra ring-theory
abstract-algebra ring-theory
edited Nov 15 at 18:19
mvw
31.2k22252
asked Nov 15 at 18:01
Seven
829
edited Nov 15 at 18:19
mvw
31.2k22252
asked Nov 15 at 18:01
Seven
829
edited Nov 15 at 18:19
mvw
31.2k22252
edited Nov 15 at 18:19
mvw
31.2k22252
edited Nov 15 at 18:19
mvw
31.2k22252
31.2k22252
asked Nov 15 at 18:01
Seven
829
asked Nov 15 at 18:01
Seven
829
asked Nov 15 at 18:01
Seven
829
829
See this question and this one. For $mathbb{Z}_n$, see this question.
– J.-E. Pin
Nov 15 at 18:15
Okay, I see there are counterexamples but they are not trivial. About $mathbb{Z}_n$, I don't see that last topic solve my question
– Seven
Nov 15 at 18:26
I got it thank you
– Seven
Nov 15 at 19:00
add a comment |
See this question and this one. For $mathbb{Z}_n$, see this question.
– J.-E. Pin
Nov 15 at 18:15
Okay, I see there are counterexamples but they are not trivial. About $mathbb{Z}_n$, I don't see that last topic solve my question
– Seven
Nov 15 at 18:26
I got it thank you
– Seven
Nov 15 at 19:00
See this question and this one. For $mathbb{Z}_n$, see this question.
– J.-E. Pin
Nov 15 at 18:15
See this question and this one. For $mathbb{Z}_n$, see this question.
– J.-E. Pin
Nov 15 at 18:15
Okay, I see there are counterexamples but they are not trivial. About $mathbb{Z}_n$, I don't see that last topic solve my question
– Seven
Nov 15 at 18:26
Okay, I see there are counterexamples but they are not trivial. About $mathbb{Z}_n$, I don't see that last topic solve my question
– Seven
Nov 15 at 18:26
I got it thank you
– Seven
Nov 15 at 19:00
I got it thank you
– Seven
Nov 15 at 19:00
add a comment |
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See this question and this one. For $mathbb{Z}_n$, see this question.
– J.-E. Pin
Nov 15 at 18:15
Okay, I see there are counterexamples but they are not trivial. About $mathbb{Z}_n$, I don't see that last topic solve my question
– Seven
Nov 15 at 18:26
I got it thank you
– Seven
Nov 15 at 19:00