Wedderburn–Malcev Decomposition

A structural theorem for finite-dimensional associative algebras stating that, under a splitting condition, an algebra A decomposes as a semidirect (or vector-space) sum A = S ⊕ R where R is the Jacobson radical and S is a semisimple subalgebra isomorphic to the semisimple quotient A/R; the decomposition provides a semisimple complement to the radical when a splitting exists.