Exercise 2, § 13

Let $A_1$ and $A_2$ be K-algebras, A the algebra $A_1\otimes_KA_2, M_i$ an $A_i$-module (for $i= 1,2$), and M the A-module $M_1\otimes_KM_2$.

a) Suppose that the $A_1$-module $M_1$ is absolutely without radical. Prove the inclusion $\mathfrak{R}_A(M)\subset M_1\otimes_K\mathfrak{R}_{A_2}(M_2)$.

b) If the modules $M_1$ and $M_2$ are absolutely without radical, then so is the A-module M.