Exercise 26, § 2

(a) Show that every Zorn ring A (Algebra, Chapter VIII, § 6, Exercise 13) with no nil ideal $\neq 0$ is left neat; in particular, every absolutely flat ring and every primitive ring whose socle is not reduced to 0 is a left neat ring.

(b) Let A be a left primitive ring whose socle is not reduced to 0; show that, if A is represented as a dense subring of a ring of endomorphisms $\operatorname{End}_D(V)$ of a vector space (Algebra, Chapter VIII, § 5, Exercise 10), the ring $A_{\mathscr{F}}$ corresponding to A (Exercise 25) is isomorphic to $\operatorname{End}_D(V)$.

(c) Show that every quasi-simple ring A (Algebra, Chapter VIII, § 5, Exercise 5) is left and right neat and that the corresponding ring $A_{\mathscr{F}}$ is quasi-simple. Deduce that, for every ring B not reduced to 0, there exists a homomorphism $\phi : B \to R$ to a ring R which is quasi-simple, absolutely flat and such that $R_s$ is an injective R-module.

(d) Let A be a left Noetherian ring with no nilpotent ideals. Show that A is left neat and that the corresponding ring $A_{\mathscr{F}}$ is a semi-simple ring ("Goldie's Theorem"; use Exercise 24 (a) and, on the other hand, use Algebra, Chapter VIII, § 2, Exercise 7, noting that in the absolutely flat ring $A_{\mathscr{F}}$ there cannot exist any family of idempotents which are pairwise orthogonal, unless A contained an infinite direct sum of left ideals $\neq 0$). Show that, if $s \in A$ is not a right divisor of 0 in A, it is invertible in $A_{\mathscr{F}}$ and conversely; the set S of these elements is a multiplicative subset of A. Show that $\mathscr{F}$ is the set of left ideals of A which meet S (to see that the ideals of A meeting S belong to $\mathscr{F}$, note that the elements of S are not left divisors of 0 in A; to see that every $f \in \mathscr{F}$ contains an element of S, represent each simple component of $A_{\mathscr{F}}$ as an endomorphism ring of an n-dimensional vector space E and define by induction n elements $u_1, \ldots, u_n$ such that $u_i u_j = 0$ for $j < i$ and $u_i^2 \neq 0$, the $u_i$ being of rank 1; finally show that the kernel of $s = u_1 + \ldots + u_n$ is zero). Deduce that $(A_{\mathscr{F}}, j_A)$ is a ring of left fractions of A with denominators in S (Exercise 22).