# Scalar extension of integral extension is integral.

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?

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 $$A$$.
• Good idea, but you got $S$ and $A$ switched up. – darij grinberg Aug 16 '19 at 11:01
• @Riquelme Also, the tensors of the form $a\oplus 1$ are integral over $S$. How do we conclude for a finite linear combination of those tesnors are also integral? – Babai Aug 17 '19 at 7:40