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)?
Best Answer
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.