Let
be a finite-dimensional associative algebra (cf.
Associative rings and algebras)
over a field
with radical
,
and let the quotient algebra
be a
separable algebra
(for algebras over a field of characteristic zero this is always true). Then
can be decomposed (as a linear space) into a direct sum of the radical
and some semi-simple subalgebra
:
and if there exists another decomposition
,
where
is a semi-simple subalgebra, then there exists an automorphism
of the algebra
which maps
onto
(the automorphism
is
inner,
i.e. there exist elements
such that
and
for all
,
where
).
The existence of this decomposition was shown by
J.H.M. Wedderburn
[1]
and the uniqueness, up to an automorphism of
the semi-simple term, was proved by
A.I. Mal'tsev
[2].
This theorem, together with Wedderburn's theorem (cf.
Associative rings and algebras)
on the structure of semi-simple algebras constitutes the central
part of the classical theory of finite-dimensional algebras.