Let $R$ be a commutative ring such that $R/J(R)$ is von Neumann regular, where $J(R)$ is the Jacobson radical of $R$. I conjecture that the idempotents of $R/J(R)$ lift modulo $J(R)$. Is it a fact?
Indeed, this is equivalent to the statement that every direct summand of the $R$-module $R/J(R)$ would have a projective cover by Proposition 27.4 of "Rings and Categories of Modules" by Fuller & Anderson. Thanks for any help!