Lifting idempotents

Remarks

Example: idempotents lift modulo $I = 4\Z/8\Z$ in $R = \Z/8\Z$. The idempotents of $R/I \cong \Z/4\Z$ are $0$ and $1$, and they lift to the idempotents $0$ and $1$ of $R$. Non-example: idempotents do not lift modulo $6\Z$ in $\Z$. The class of $3$ is idempotent in $\Z/6\Z$, since $9 - 3 = 6$, but the idempotents of $\Z$ are $0$ and $1$, and neither $3 - 0$ nor $3 - 1$ lies in $6\Z$. Idempotents lift modulo every nil ideal, and modulo every ideal of a ring that is complete with respect to the $I$-adic topology. A semilocal ring in which idempotents lift modulo the Jacobson radical is called semiperfect.