Let $C\subset B\subset A$ be rings. If $B$ is integral over $C$ and $A$ is integral over $B$ , then $A$ is integral over $C$ .

Proof. Choose $u\in A$ . Then $u^n+b_1 u^{n-1}+\cdots+b_n=0, b_i\in B$ . Thus $C[b_1,\ldots,b_n,u]$ is integral and thus module-finite over $C[b_1,\ldots,b_n]$ . Each $b_i$ is integral over $C$ , so $C[b_1,\ldots,b_n]$ is integral hence module-finite over $C$ . Thus $C[b_1,\ldots,b_n,u]$ is module-finite, hence integral, over $C$ , so $u$ is integral over $C$ .