If {"Ai"} is a finite collection of simple algebras, then their Cartesian product ∏ "Ai" is semisimple. If ("ai") is an element of Rad("A") and "e"1 is the multiplicative identity in "A"1 (all simple algebras possess a multiplicative identity), then ("a"1, "a"2, ...) · ("e"1, 0, ...) = ("a"1, 0..., 0) lies in some nilpotent ideal of ∏ "Ai". This implies, for all "b" in "A"1, "a"1"b" is nilpotent in "A"1, i.e. "a"1 ∈ Rad("A"1). So "a"1 = 0. Similarly, "ai" = 0 for all other "i".