Exercise 11, § 4

Let L be a commutative ring, A a left Artinian algebra over L and $\gamma$ a mapping of $A \times A$ into L. Define on A the internal law of composition $(a, b) \mapsto a * b = ab + \gamma(a, b)ba$. For every subset E of A, let $\tilde{E}$ denote the subring (with no unit element) of A generated by E. Show that if E consists of nilpotent elements and is stable under the law *, then $\tilde{E}$ is nilpotent (that is, there exists $n > 0$ such that $\tilde{E}^n = \{0\}$). Proceed as follows:

1. Suppose first that A is a simple ring and hence isomorphic to $\mathscr{L}_D(T)$, where T is a left vector space of finite dimension m over a field D. Argue by induction on m. Let $\Phi$ be the set of subsets F $\subset$ E which are stable under \* and such that $\tilde{F}$ is nilpotent. Show that $\Phi$ admits a maximal element M (note that $\tilde{F}^m = \{0\}$ for all $F \in \Phi$). Suppose $\tilde{M} \neq \tilde{E}$. Show that there exists $a \in E$ such that $a \notin \tilde{M}$ and $t * a \in \tilde{M}$ for all $t \in M$ (observe that there can be no infinite sequence $(a_n)$ such that $a_n \in E$, $a_n \notin M$, $a_n = t_{n-1} * a_{n-1}$ with $t_{n-1} \in M$). Let S be the subspace of T the direct sum of the $u(T)$ with $u \in \tilde{M}$; show that $S \neq \{0\}$ and $S \neq T$ and that $a(S) \subset S$. Let N be the set of $v \in E$ such that $v(S) \subset S$. By using the induction hypothesis and considering the elements of N as operating on S and on T/S, show that $N \in \Phi$, which implies a contradiction.

2. In the general case, use the fact that the Jacobson radical of A is nilpotent. Deduce from this result a new proof of Engel’s Theorem.