Exercise 5, § 10

Let $n$ be an integer. A ring A is called $n$-regular if for every element $a$ of A, there exists an $x\in A$ such that $xa^{n+1}=a^n$.

a) Suppose that A is $n$-regular. Prove that we have $r^n= 0$ for every element $r$ of $\mathfrak{R}(A)$.

b) Prove that an $n$-regular primitive ring is isomorphic to a matrix ring $\mathbf{M}_r(D)$ over a field, with $r\leqslant n$. (Observe that if V is a D-vector space of dimension $> n$ and $e$ a nonzero vector in V, then every dense subring of End$_D(V)$ contains an element $u$ such that $u^n(e)\not= 0$ and $u^{n+1}(e) = 0.$)

c) Prove that an $n$-regular ring is a Zorn ring (VIII, p. 171, Exercise 26). (First, observe that the relation $xa^{n+1}=a^n$ implies $x^na^{2n}=a^n$. Using b), Exercise 12, a) of VIII, p. 168 and Proposition 2 of VIII, p. 27, deduce that we have $a^nx^na^n-a^n\in$ $\mathfrak{R}(A)$. Finally, using Lemma 1 of VIII, p. 159, prove that if two elements $y$ and $z$ of A satisfy $y=zy^2$ and $yzy-y\in \mathfrak{R}$(A), then there exists a $t\in A$ such that $y=ty^2$ and $(ty)^2=ty.)$ If the ring A is, moreover, without radical, then it is pseudoregular (Exercise 4).

d) Let A be a ring with no nilpotent elements other than 0. Then A is $n$-regular if and only if it is absolutely flat.

(If A is absolutely flat, use Exercise 27, b) of VIII, p. 171. If A is $n$-regular, first observe, with the notation above, that $e=x^na^n$ is idempotent, hence central (Exercise 30, a) of VIII, p. 172), and that we have $ea=a$, so that A is 1-regular. Then use questions a) and b) to prove that A is isomorphic to a subring of a product of fields and deduce that the relation $xa^{n+1}=a^n$ implies $axa=a.)$