Scalar extension of integral extension is integral.

Question

Let $A$ be an $R$-algebra which is integral over $R$ and $f:R\rightarrow S$ a ring homomorphism. Show that the scalar extensions $S\otimes _RA$ is integral over $A$.

Can you someone give any hint?

Answer 1

Let $s_i \in S$ be generators for $S$ over $R$. By assumption each of the $s_i$ satisfies a monic equation $f_i$ over $R$. But then the elements $s_i \otimes 1 \in S\otimes _RA$ generate $S\otimes _RA$ over $A$ AND satisfy the corresponding polynomial eq. $f^A_i$ over $A$, so $S\otimes _RA$ is integral over $A$.