Exercise 4, § 19

Let $(F, s)$ be an Artinian (associative) Cayley K-algebra, and let $\mathfrak{r}$ be its radical. a) Prove that we have $x^2= 0$ and $s(x) =-x$ for every $x\in \mathfrak{r}$; the algebra $\overline{F}= F/\mathfrak{r}$ is Cayley and isomorphic to one of the algebras described in Exercise 3.

b) Prove that if $\overline{F}$ is a quaternion algebra over K, then $\mathfrak{r}$ is reduced to 0 (observe that we have $(ab-ba)x= 0$ for $a, b\in F,x\in \mathfrak{r}$). c) Suppose that $\overline{F}$ is commutative. Prove that $\mathfrak{r}^3$ is reduced to 0 but that $\mathfrak{r}^2$ may not be.

d) Suppose that $\overline{F}$ is commutative and $\mathfrak{r}^2$ is reduced to 0. Prove that there exists an F-module V such that F is isomorphic to $\overline{F}\oplus V$ endowed with the structure of a K-algebra for which we have $(\lambda , v)(\mu, w) = (\lambda \mu, \lambda w+s(\mu)v)$. (In cases $\alpha$ ) through $\gamma )$ of Exercise 3, c), use Corollary 1 of VIII, p. 245. In case $\delta )$, choose a $p$-basis $(e_i)_{i\in I}$ of $\overline{F}$ over K and for each $i\in I$, an element $f_i$ of F that lifts $e_i$, and prove that the subalgebra of F generated by the $(f_i)$ is isomorphic to F.)