I am learning about ideals in my algebra class. If I have a ring $R$, I know that two ideals $I$ and $J$ in $R$ are coprime if $I+J=R$. I also know that $\mathbb{Z}$ is a principal ideal domain. I was TOLD that if you have two integers that are coprime then the ideals that they generate are coprime.

However, I know that $1 \in \mathbb{Z}$ is a unit. And I know that if you have a unit in your ideal then you end up generating the whole ring.

So actually my question has to do with a general ring: how is it possible to have two ideals be coprime (and not trivial)?


If $R$ is a local commutative ring with $1$, then it has not coprime ideals. If $R$ has two maximal ideals $m$, $n$, then $m$ and $n$ are coprime.

In $\mathbb{Z}$, if different prime numbers appear in the decomposition of two integers, then the ideals generated by them are coprime.

  • $\begingroup$ So does that mean that what I was told about coprime integers is incorrect? $\endgroup$ – Rebekah Feb 16 '15 at 7:14
  • $\begingroup$ no, Z is not local. $\endgroup$ – user 1 Feb 16 '15 at 7:15
  • $\begingroup$ I don't understand how $(2)+(3)$ could generate $\mathbb{Z}$. For example, how do you get 5? $\endgroup$ – Rebekah Feb 16 '15 at 7:21
  • $\begingroup$ (2) means all integers 2k and similarly for (3). and 5=2+3 $\endgroup$ – user 1 Feb 16 '15 at 7:23
  • 1
    $\begingroup$ in fact since $1= 3+(-1)2$, so $1\in (2)+(3)$. therefore every integer belong to $(2)+(3)$ $\endgroup$ – user 1 Feb 16 '15 at 7:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.