# Wedderburn-Mal'tsev theorem

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Let $A$ be a finite-dimensional associative algebra (cf. Associative rings and algebras) over a field $F$ with radical $N$, and let the quotient algebra $A/N$ be a separable algebra (for algebras over a field of characteristic zero this is always true). Then $A$ can be decomposed (as a linear space) into a direct sum of the radical $N$ and some semi-simple subalgebra $S$:

$$A = N \oplus S,$$

and if there exists another decomposition $A = N \oplus {S _ {1} }$, where $S _ {1}$ is a semi-simple subalgebra, then there exists an automorphism $\phi$ of the algebra $A$ which maps $S$ onto $S _ {1}$( the automorphism $\phi$ is inner, i.e. there exist elements $a, a ^ \prime \in A$ such that $a \cdot a ^ \prime = a ^ \prime \cdot a = 0$ and $x \phi = a \cdot x \cdot a ^ \prime$ for all $x \in A$, where $x \cdot y = x + y + xy$). The existence of this decomposition was shown by J.H.M. Wedderburn  and the uniqueness, up to an automorphism of the semi-simple term, was proved by A.I. Mal'tsev . 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.

How to Cite This Entry:
Wedderburn–Mal'tsev theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wedderburn%E2%80%93Mal%27tsev_theorem&oldid=23128