Let $R$ be a commutative ring. Using the definition that two ideals $I, J \subseteq R$ are relatively prime if $I + J = R$.
I want to show that for two relatively prime ideals $I, J \subseteq R$, it holds true that:
$I \cap J = I J$, and that the transformation: $f: R/I J \to R/I \oplus R/J$ is an isomorphism.
Thanks in advance. I discovered another thread (Intersection of ideals generated by two relatively prime elements) where the first statement was dealt with in case that R is a principal ideal domain and the two ideals therefore generated by only one element. I'm not sure how to prove it generally, with R just being any commutative ring and the ideals not necessarily being principal.