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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?

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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 $A$.

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    $\begingroup$ Good idea, but you got $S$ and $A$ switched up. $\endgroup$ – darij grinberg Aug 16 '19 at 11:01
  • $\begingroup$ Yes, the S and A are interchanged. $\endgroup$ – Babai Aug 17 '19 at 6:43
  • $\begingroup$ @Riquelme Why should there be finitely many generators of A over R? $\endgroup$ – Babai Aug 17 '19 at 7:26
  • $\begingroup$ @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? $\endgroup$ – Babai Aug 17 '19 at 7:40

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