# Lifting idempotents modulo $J(R)$

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!

You can semi localize $\mathbb Z$ at two maximal primes, so that the ring mod the radical is a product of two fields.
• Thanks for your cooperation! What do you mean by "... semi localize $\mathbb Z$ at two maximal primes"? Principally, what does mean "to semi localize"? – karparvar Dec 23 '16 at 17:06
• @karparvar I think is pretty clear from the answer what this means: localize a ring $R$ at $R\setminus(p_1\cup p_2)$ for two prime ideals $p_1,p_2$. – user26857 Dec 24 '16 at 0:13