Corollary 2

Let $(M_i)_{i\in I}$ be a family of A-modules, P its product, and S its direct sum. We have $\mathfrak{R}(P)\subset \prod_{i\in I}\mathfrak{R}(M_i)$ and $\mathfrak{R}(S) =\bigoplus_{i\in I}\mathfrak{R}(M_i)$.

For $i\in I$, let $\pi_i$ be the projection with index $i$ from P to $M_i$. By Proposition 1, we have $\pi_i(\mathfrak{R}(P))\subset \mathfrak{R}(M_i)$ for every $i\in I$; this proves the first assertion. We have $S\subset P$, hence

Moreover, we have $M_i\subset S$ and therefore $\mathfrak{R}(M_i)\subset \mathfrak{R}(S)$ for every $i\in I$, and consequently $\oplus_{i\in I}\mathfrak{R}(M_i)\subset \mathfrak{R}(S)$.

There exist families of modules such that the radical of the product is not isomorphic to the product of the radicals (Exercise 3 of VIII, p. 166).