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An Algebra with no nontrivial nilpotent Ideals. In the 1890s, Cartan, Frobenius, and Molien independently proved that any finite-dimensional semisimple algebra over the Real or Complex numbers is a finite and unique Direct Sum of Simple Algebras. This result was then extended to algebras over arbitrary fields by Wedderburn in 1907 (Kleiner 1996).
See also Ideal, Nilpotent Element, Simple Algebra
References
Kleiner, I. ``The Genesis of the Abstract Ring Concept.'' Amer. Math. Monthly 103, 417-424, 1996.