Corollary 3

The radical of the product of a family $(A_i)_{i\in I}$ of rings is the product of the $\mathfrak{R}(A_i)$.

Let $x= (x_i)_{i\in I}$ be an element of $\prod_{i\in I}A_i$. For every element $a= (a_i)_{i\in I}$ of $\prod_{i\in I}A_i$, the element $1 +ax$ is left invertible if and only if $1 +a_ix_i$ is left invertible in $A_i$ for every $i\in I$. Corollary 3 follows.