An algebra "A" is called "simple" if it has no proper ideals and "A"2 = {"ab" | "a", "b" ∈ "A"} ≠ {0}. As the terminology suggests, simple algebras are semisimple. The only possible ideals of a simple algebra "A" are "A" and {0}. Thus if "A" is simple, then "A" is not nilpotent. Because "A"2 is an ideal of "A" and "A" is simple, "A"2 = "A". By induction, "An" = "A" for every positive integer "n", i.e. "A" is not nilpotent.