Why is the definition of coprime/comaximal ideals $I + J = R$?

Question

Two questions

Let $R$ be commutative and have $1_R$. Two ideals are called coprime/comaxmal if $I + J = R$

(1) the above is equivalent to saying there exists $i + j = 1_R$. But to me, this condition should be definition and everywhere (i.e. Chinese Reminder) I look, they use the element condition, never $I + J = R$. So why don't we just say coprime/comaximal if $i + j = 1_R$?

(2) I understand why it is called co-prime, but what does co-maximal even mean? Co means together, and what does maximal have to do with this? We aren't on a PID.

Answer 1

You often want your definition to be as general as possible:

For any ring $R$, we'll say two ideals $I$ and $J$ are comaximal if $I+J=R$. In particular, if $R$ is unital with identity $1_R$, then these ideals are comaximal if there exists $i\in I$ and $j \in J$ such that $i+j = 1_R$.

One definition that works for any ring is cleaner than having two definitions, one in the case that your ring is unital and one for non-unital rings.

I made this Community Wiki so someone else can explain why "comaximal" is a good choice of word.