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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)?

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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.

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  • $\begingroup$ 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
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    $\begingroup$ 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

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