Exercise 30, § 9

a) Let A be a ring that contains no nilpotent elements other than 0. Prove that every idempotent $e$ in A belongs to the center of A (consider the elements $(1-e)xe$ and $ex(1-e)$).

b) Let A be an absolutely flat ring that contains no nilpotent elements other than 0, and let $a$ be a nonzero element of A. Prove that there exists an element $x$ of A such that $e=ax=xa$ is idempotent and $ea=ae=a$. Deduce that for every $y\in A$, there exists a $z\in A$ such that $ya=az$ and, consequently, that every left or right ideal of A is two-sided.

c) Prove that no quotient ring of A contains a nonzero nilpotent element (use b)). Deduce that A is isomorphic to a subring C of a product $\prod_{\iota\in I}D_{\iota}$ of fields such that pr$_{\iota}(C) = D_{\iota}$ for every $\iota \in I$ (deduce from b) and Exercise 4 of VIII, p. 129 that an absolutely flat primitive ring without nilpotent elements is a field, and use Exercise 12 above).

d) Let $(D_{\iota})_{\iota\in I}$ be an infinite family of fields and B the ring $\prod_{\iota\in I}D_{\iota}$. For any subset H of I, denote by $B_H$ the two-sided ideal of B consisting of the families $(b_{\iota})_{\iota\in I}$ such that $b_{\iota}= 0$ for $\iota \notin H$. Let $\mathfrak{F}$ be an increasing directed set of subsets of I forming a cover of I, and let A be the subring of B generated by 1 and the $B_H$ for $H\in \mathfrak{F}$. Prove that A is absolutely flat and contains no nonzero nilpotent elements.