Theorem 1

Let K be a commutative field and A a K-algebra. The following properties are equivalent:

(i) The K-algebra A is absolutely semisimple.

(ii) The algebra A has finite degree over K, and there exists an extension P of K that is a perfect field, for which the P-algebra $A_{(P)}$ is semisimple.

(iii) The K-algebra A is semisimple and has finite degree over K, and its center is an étale K-algebra.

(iv) There exists a finite family $(n_i,D_i)_{i\in I}$, where $n_i$ is a strictly positive integer and $D_i$ a K-algebra of finite degree that is a field, such that the center $Z_i$ of $D_i$ is a separable extension of K for every $i\in I$ and A is isomorphic to the product of the matrix rings $\mathbf{M}_{n_i}(D_i)$.

(v) There exist an extension L of K and a finite family of integers $(n_i)_{i\in I}$ such that the L-algebra $A_{(L)}$ is isomorphic to the algebra $\prod_{i\in I}\mathbf{M}_{n_i}(L)$.

(vi) There exist a Galois extension L of K of finite degree and a finite family of integers $(n_i)$ such that $A_{(L)}$ is isomorphic to the product of the matrix algebras $\mathbf{M}_{n_i}(L)$.

We first prove the implications (v) $\Rightarrow$ (iv) $\Rightarrow$ (iii) $\Rightarrow$ (ii) $\Rightarrow$ (i). Denote the center of A by Z.

If property (v) holds, then the L-algebra $A_{(L)}$ is semisimple and has finite degree over L, and its center (isomorphic to $Z_{(L)}$, III, p. 41, Corollary) is isomorphic to $L^r$ for some integer $r\geqslant 0$. By Corollary 2, a) of VIII, p. 222, the algebra A is semisimple and has finite degree over K. It is therefore isomorphic to a finite product of rings $\prod_{i\in I}\mathbf{M}_{n_i}(D_i)$ with $n_i\geqslant 1$ for every $i\in I$, where the field $D_i$ is an algebra of finite degree over K. The center $Z_i$ of $D_i$ is a commutative field that is an extension of K, and Z is isomorphic to $\prod_{i\in I}Z_i$. Therefore, $Z_{(L)}$ is isomorphic, on the one hand, to $\prod_{i\in I}(Z_i)_{(L)}$ and, on the other hand, to $L^r$. In other words, the algebra $\prod_{i\in I}Z_i$ is étale over the field K (V, §6, No. 3, p. 28, Definition 1), and each of the extensions $Z_i$ is separable over K (V, §6, No. 4, p. 31, Corollary, and V, §7, No. 1, p. 42). So (v) implies (iv).

If property (iv) holds, then it is clear that A is a semisimple algebra and has finite degree over K. Its center Z is isomorphic to the product $\prod_{i\in I}Z_i$ of separable extensions of K of finite degree; it is therefore an étale algebra (loc. cit.). So (iv) implies (iii).

The implications (iii) $\Rightarrow$ (ii) $\Rightarrow$ (i) follow from Proposition 2 of VIII, p. 230 applied to the A-module $A_s$.