Example of commutative ring with two elements that don't generate entire ring
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I believe that there should be an example of a commutative ring $R$ that contains two elements whose only
common divisors are units but which do not generate the unit ideal.
$R$ can't be a Euclidean domain. I've been trying out some basic examples of non-integral domains, such as $mathbbZ/4mathbbZ$, but without luck so far.
Is there such an example?
abstract-algebra ring-theory examples-counterexamples
add a comment |Â
up vote
1
down vote
favorite
I believe that there should be an example of a commutative ring $R$ that contains two elements whose only
common divisors are units but which do not generate the unit ideal.
$R$ can't be a Euclidean domain. I've been trying out some basic examples of non-integral domains, such as $mathbbZ/4mathbbZ$, but without luck so far.
Is there such an example?
abstract-algebra ring-theory examples-counterexamples
2
How about $Bbb Z[X,Y]$?
â Lord Shark the Unknown
Aug 13 at 6:41
You won't be able to make this work if all ideals are principal.
â Mark Bennet
Aug 13 at 7:25
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I believe that there should be an example of a commutative ring $R$ that contains two elements whose only
common divisors are units but which do not generate the unit ideal.
$R$ can't be a Euclidean domain. I've been trying out some basic examples of non-integral domains, such as $mathbbZ/4mathbbZ$, but without luck so far.
Is there such an example?
abstract-algebra ring-theory examples-counterexamples
I believe that there should be an example of a commutative ring $R$ that contains two elements whose only
common divisors are units but which do not generate the unit ideal.
$R$ can't be a Euclidean domain. I've been trying out some basic examples of non-integral domains, such as $mathbbZ/4mathbbZ$, but without luck so far.
Is there such an example?
abstract-algebra ring-theory examples-counterexamples
asked Aug 13 at 6:30
CuriousKid7
1,531617
1,531617
2
How about $Bbb Z[X,Y]$?
â Lord Shark the Unknown
Aug 13 at 6:41
You won't be able to make this work if all ideals are principal.
â Mark Bennet
Aug 13 at 7:25
add a comment |Â
2
How about $Bbb Z[X,Y]$?
â Lord Shark the Unknown
Aug 13 at 6:41
You won't be able to make this work if all ideals are principal.
â Mark Bennet
Aug 13 at 7:25
2
2
How about $Bbb Z[X,Y]$?
â Lord Shark the Unknown
Aug 13 at 6:41
How about $Bbb Z[X,Y]$?
â Lord Shark the Unknown
Aug 13 at 6:41
You won't be able to make this work if all ideals are principal.
â Mark Bennet
Aug 13 at 7:25
You won't be able to make this work if all ideals are principal.
â Mark Bennet
Aug 13 at 7:25
add a comment |Â
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2
How about $Bbb Z[X,Y]$?
â Lord Shark the Unknown
Aug 13 at 6:41
You won't be able to make this work if all ideals are principal.
â Mark Bennet
Aug 13 at 7:25