Exercise 18, § 9

a) Let A be a ring, $\mathfrak{a}$ a two-sided nil ideal of A, and $(e_n)_{n\geqslant 1}$ an orthogonal sequence of idempotents in $A/\mathfrak{a}$. Prove that there exists an orthogonal sequence $(e_n)_{n\geqslant 1}$ of idempotents in A such that $e_n$ is the class of $e_n$ in $A/\mathfrak{a}$ for every $n$. (Let $e$ and $e'$ be two orthogonal idempotents in $A/\mathfrak{a},e'$ an idempotent in A lifting $e'$, and let $a$ be a representative of $e$ in A. Observe that we can modify $a$ to have $ae'=e'a$, and then, as in Proposition 7 of VIII, p. 159, to have $a^2=a.$) b) Let K be a commutative field, L an uncountable set, and A the K-algebra defined in the previous exercise. Prove that there does not exist any orthogonal family $(e_{\lambda})_{\lambda\in L}$ of idempotents in A satisfying $e_{\lambda}\equiv c_{\lambda}$ (mod $\mathfrak{R}(A)$) for every $\lambda \in L$. (Consider the set H of elements $(\lambda , \mu)$ of $L\times L$ such that $c_{\lambda}(e_\mu-c_\mu) = 0$. Observe that the sets $H\cap (\{\lambda \} \times L)$ are finite for every $\lambda \in L$ and the sets $H\cap (L\times \{\mu\})$ are finite for every $\mu\in L.$)