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?







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  • 2




    How about $Bbb Z[X,Y]$?
    – Lord Shark the Unknown
    Aug 13 at 6:41










  • You won't be able to make this work if all ideals are principal.
    – Mark Bennet
    Aug 13 at 7:25














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?







share|cite|improve this question
















  • 2




    How about $Bbb Z[X,Y]$?
    – Lord Shark the Unknown
    Aug 13 at 6:41










  • You won't be able to make this work if all ideals are principal.
    – Mark Bennet
    Aug 13 at 7:25












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?







share|cite|improve this question












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?









share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Aug 13 at 6:30









CuriousKid7

1,531617




1,531617







  • 2




    How about $Bbb Z[X,Y]$?
    – Lord Shark the Unknown
    Aug 13 at 6:41










  • You won't be able to make this work if all ideals are principal.
    – Mark Bennet
    Aug 13 at 7:25












  • 2




    How about $Bbb Z[X,Y]$?
    – Lord Shark the Unknown
    Aug 13 at 6:41










  • You won't be able to make this work if all ideals are principal.
    – Mark Bennet
    Aug 13 at 7:25







2




2




How about $Bbb Z[X,Y]$?
– Lord Shark the Unknown
Aug 13 at 6:41




How about $Bbb Z[X,Y]$?
– Lord Shark the Unknown
Aug 13 at 6:41












You won't be able to make this work if all ideals are principal.
– Mark Bennet
Aug 13 at 7:25




You won't be able to make this work if all ideals are principal.
– Mark Bennet
Aug 13 at 7:25















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