Corollary

Let A be a commutative ring and B an algebra over A. Suppose that either the ring A is Artinian or B is a finitely generated A-module. We then have $\mathfrak{R}(A)B\subset \mathfrak{R}(B)$.

By Proposition 3, the radical $\mathfrak{R}_A(B_s)$ is contained in $\mathfrak{R}_B(B_s)$, which is nothing but the radical of the ring B. Moreover, we have $\mathfrak{R}(A)B\subset \mathfrak{R}_A(B_s)$ by Proposition 6 of VIII, p. 158. The corollary follows.