# Coprime ideal definition

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.
• I don't understand how $(2)+(3)$ could generate $\mathbb{Z}$. For example, how do you get 5? – Rebekah Feb 16 '15 at 7:21
• in fact since $1= 3+(-1)2$, so $1\in (2)+(3)$. therefore every integer belong to $(2)+(3)$ – user 1 Feb 16 '15 at 7:27