The chinese remainder theorem can be generalized as follows: Let $G$ be a group with normal subgroups $H,K$ of $G$. Then the canonical map $G/(H \cap K) \to G/H \times_{G/(HK)} G/K$ is an isomorphism. The proof is trivial! With the same argument: Let $R$ be a ring with ideals $I,J$, then the canonical map $R/(I \cap J) \to R/I \times_{R/(I+J)} R/J$ is an isomorphism. Now this reveals a geometric meaning of the cinese remainder theorem:
Let $X$ be a scheme and $A,B$ two closed subschemes of $X$. Then $A \cup B$ is a closed subscheme of $X$ (intersect the ideal sheaves) and we have $A \cup B = A \coprod_{A \cap B} B$.
Thus the union of two closed subschemes is the pushout of $A$ and $B$ along $A \cap B$, which is very intuitive.