Exercise 16, § 5

Let $g$ be a Lie algebra over an arbitrary field $K$, $A$ a commutative associative algebra over $K$ and $g' = g \otimes_K A$, which can be considered as a Lie algebra over $K$.

(a) If $D$ is a derivation of $A$, show that there exists one and only one derivation $D'$ of $g'$ such that $D'(x \otimes a) = x \otimes Da$ for $x \in g, a \in A$.

(b) Let $p$ be a prime number, $G$ a cyclic group of order $p$ and $s$ a generator of $G$. Suppose that $K$ is of characteristic $p$ and henceforth let $A$ be the algebra of $G$ over $K$. Show that there exists one and only one derivation $D$ of $A$ such that $D(s^k) = ks^{k-1}$ for all $k \in \mathbf{Z}$. Show that the $K$-linear combinations of the $x \otimes (s-1)^k$ ($k = 1, 2, \ldots, p-1, x \in g$) form a solvable ideal $r$ of $g'$ and that $g'/r$ is isomorphic to $g$.

(c) Take $g$ to be simple (cf. § 6, no. 2, Definition 2). Then $r$ is the radical of $g'$ but is not a characteristic ideal. (Observe that $D(x \otimes (s-1)) = 1 \otimes x$.)[^1]

[^1]: This result, unedited, was communicated to us by N. Jacobson.