Exercise 4, § 8

Let A be a semiprime left Noetherian ring (Exercise 3) and S the set of its cancelable elements (cf. VIII, p. 75, Exercise 8).

a) Prove that every nonzero (left or right) ideal of A contains an element that is not nilpotent. (Let $\mathfrak{a}$ be a right ideal of A consisting of nilpotent elements. Let $x$ be a nonzero element of $\mathfrak{a}$ whose left annihilator is maximal. Observe that for every element $a$ of A and every integer $n\geqslant 1$ such that $(xa)^n\not= 0$, the left annihilator of $(xa)^n$ is equal to that of $x$, and deduce that we have $xax= 0.$)

b) Prove that S allows a calculus of left fractions on A (VIII, p. 18, Exercise 18). (For $a\in A$ and $s\in S$, prove that the set of elements $x$ of A such that $xa\in As$ is an irreducible left ideal of A (VIII, p. 75, Exercise 7), and conclude using a) and Exercise 8, d) of VIII, p. 75.)

c) Prove that the ring $S^{-1}A$ is semisimple (“Goldie’s theorem” [^1]). (Let $\mathfrak{b}$ be an irreducible left ideal of $S^{-1}A$. Show that $A\cap \mathfrak{b}$ is an irreducible left ideal of A, and then use Exercise 8, d) of VIII, p. 75 to show that $\mathfrak{b}= S^{-1}A$. Conclude using Exercise 7, b) of VIII, p. 75.)

[^1]: For other approaches to this result, see X, §1, p. 171, exercice 25 and Comm. Alg., II, §2, p. 135, Exercise 26.