I am working on a project on generalization of the Chinese Remainder Theorem in commutative rings, which inevitably have to go through the definition of coprimality in commutative rings. I came across page 65 of this paper here which says:
Definition 2.12: The ideals $\mathscr I$ and $\mathscr J$ of $R$ a commutative ring are relatively prime if $\mathscr I + \mathscr J = R.$
Honestly, I don't feel comfortable present it as it is on my paper without any narration explaining the rationale behind it. It looks so intuitive but I am lost on how to elaborate it, I have been searching left and right for rationales but could not find any. Any links or pointers would be very much appreciated.
Thank you for your time and help.
Best Answer
This is essentially a generalisation of the notion of coprimality in the integers. If two integers are coprime, then their gcd is 1 (and by Bézout's identity gcd can be expressed as a linear combination of those two numbers), which is the generator of the entire ring $Z$. Hence the sum of their ideals is $Z$. So while I can't say what to expect in a ring without a multiplicative identity, i.e 1, but in the rings with 1 this seems to be the best way to define coprimality.