There's a small detail in a proof of the Chinese Remainder Theorem for modules I don't understand when it comes to showing the normally constructed homomorphism is a surjection.
Suppose $R$ is a commutative unital ring, with $I_1,\dots,I_k$ pairwise comaximal ideals. How then for any $j$ can I find $r\in R$ such that $r\equiv 1\pmod{I_j}$ but $r\equiv 0\pmod{I_i}$ for $i\neq j$?
Thank you!