Exercise 27, § 9

A ring A is called absolutely flat, or von Neumann regular, if for every element $a$ of A, there exists an $x\in A$ such that $axa=a$.

a) A ring A is absolutely flat if and only if every monogenous left (or right) ideal of A admits a supplement in $A_s$ (that is, is generated by an idempotent).

b) Show that every finitely generated ideal of an absolutely flat ring is generated by an idempotent. (Reduce to the case of an ideal $Ae+ Af$, where $e$ and $f$ are idempotents. Then, by replacing $Ae$ with $Ae(1-f)$, reduce to the case when $ef= 0$, and consider the element $e+f-f e.$) $A$ (left or right) Noetherian absolutely flat ring is semisimple.

c) Deduce from b) that the intersection of two monogenous left (resp. right) ideals of an absolutely flat ring is monogenous (consider the right annihilators).

d) Every quotient ring of an absolutely flat ring is absolutely flat, as is every product of absolutely flat rings.

e) Prove that the radical of an absolutely flat ring is reduced to 0. Deduce that every two-sided ideal of A is the intersection of the maximal left ideals containing it. f ) The center of an absolutely flat ring is absolutely flat (prove that if $a\in Z$ and $x\in A$ satisfy $a^2x=a$, then we have $a^2x^n\in Z$ for every $n\geqslant 0$ and $a(a^2x^3)a=a$). g) Prove that the endomorphism ring of a vector space is an absolutely flat ring.