Exercise 7, § 1

Let A be a ring (possibly without unit element) such that the relation $2a = 0$ in A implies $a = 0$. Let U be a subring of A and an ideal of the Lie $\mathbf{Z}$-algebra associated with A.

(a) If $x, y$ belong to U, then $(xy - yx)A \subset U$ (write:

and $A(xy - yx)A \subset U$ (write:

(b) Suppose that U is commutative. Let $x \in U, s \in A$ and $y = xs - sx$. Show that $(yt)^3 = 0$ for all $t \in A$. (The element x commutes with $xs - sx$ and $xs^2 - s^2x$; deduce that $2(xs - sx)^2 = 0$, whence $y^2 = 0$. Similarly $(yt - ty)^2 = 0$ for all $t \in A$.)

(c) Suppose henceforth that the only two-sided ideals of A are {0} and A and that for every non-zero element $y$ of A there exists $t \in A$ such that $yt$ is not nilpotent. Show that, if $U \neq A$, U is contained in the centre of A. (Show, using (a), that U is commutative and then use (b).)

(d) Let V be an ideal of the Lie $\mathbf{Z}$-algebra associated with A. Then either V is contained in the centre of A or $V \supset [A, A]$. (Apply (c) to the set U of $t \in A$ such that $[t, A] \subset V$; use the identity $[xy, z] = [x, yz] + [y, zx]$.)