Let $\varphi\colon R\to(R/I)\oplus(R/J)$ where $\varphi(r) = (r + I, r + J)$. Show $\varphi$ is an isomorphism iff $I + J = R$ and $I \cap J = \{0\}$.

Question

Let $R$ be a ring and $I$, $J$ be two ideals of $R$. Let $\varphi\colon R\to(R/I)\oplus(R/J)$ where $\varphi(r) = (r + I, r + J)$. Show $\varphi$ is an isomorphism iff $I + J = R$ and $I \cap J = \{0\}$.

I need to prove the statement above. I haven't yet studied ring isomorphisms but I have general idea about isomorphisms and ideals. I can't understand the first side of the prove, like if I assumed $\varphi$ was an isomorphism then how to prove the second part? Any hints regarding explaining the idea of even proofs will be great !

Answer 1

If $\varphi$ is an isomorphism, then we get for $r \in I \cap J$ that $r \in \ker(\varphi)=0$. So we have $I \cap J=0$.

Since $\varphi$ is surjective, we have an element $r \in R$ such that $\varphi(r)=(1,0)$. This means that $r \in J$ and $1-r \in I$. Therefore we conclude: $1=(1-r)+r \in I+J$, so $R=I+J$.