Exercise 21, § 2

Let S be a commutative ring, R a commutative S-algebra, B an S-algebra (commutative or otherwise) and $B_{(R)}$ the R-algebra obtained from B by extension of scalars. Suppose that R is a flat S-module and that B is a finitely generated S-module. If Z is the centre of B, show that the canonical homomorphism of $Z_{(R)} = Z \otimes_S R$ to $B_{(R)}$ is an isomorphism of $Z_{(R)}$ onto the centre of $B_{(R)}$. (Use the exact sequence

where $\theta(x)(y) = xy - yx$, and Proposition 11 of no. 10.)