Answer to Question #17356 in Abstract Algebra for sanches

Question

Show that, for any direct product of rings Ri, rad ((direct product)Ri) = (direct product) rad Ri.

Answer 1

Let y = (yi) ∈(product)Ri. Since y ∈rad ((product)Ri) amounts to 1 − xy being left-invertible for any x = (x_i) ∈ (product)Ri. This, in turn, amounts to 1 − x_iy_i being left-invertible in Ri, for any x_i ∈ Ri (and any i). Therefore, y ∈rad ((product) Ri) iff y_i ∈rad Ri for all i.