It is less apparent from the definition that the converse of the above is also true, that is, any finite-dimensional semisimple algebra is isomorphic to a Cartesian product of a finite number of simple algebras. The following is a semisimple algebra that appears not to be of this form. Let "A" be an algebra with Rad("A") ≠ "A". The quotient algebra "B" = "A" ⁄ Rad("A") is semisimple: If "J" is a nonzero nilpotent ideal in "B", then its preimage under the natural projection map is a nilpotent ideal in "A" which is strictly larger than Rad("A"), a contradiction.