Exercise 3, § 9

a) Let A be a ring with semisimple quotient ring $A/\mathfrak{R}(A)$. Prove that we have $\mathfrak{R}(M) =\mathfrak{R}(A)M$ for every A-module M (observe that $M/\mathfrak{R}(A)M$ is an $A/\mathfrak{R}(A)$-module).

b) Deduce an example of a ring A and a family $(V_i)_{i\in I}$ of A-modules such that $\mathfrak{R}(\prod M_i)\not=\prod\mathfrak{R}(M_i)$ (use a) for a ring whose radical is not finitely generated).