A ring $R$ is said to be semiprimary if $R/J(R)$ is semisimple and $J(R)$ is nilpotent, and $R$ is said to be semiperfect if $R/J(R)$ is semisimple and for every idempotent $x\in R/J(R)$ there exists an idempotent $a\in R$ such that $x=\overline{a}$. It is possible to show that every semiprimary ring is semiperfect.
I am looking for an example of a semiperfect ring that is not semiprimary, but I can't find one. Does anyone know such an example?
For example, given a field $k$, the ring of formal power series $R=k[[x]]$ is semiperfect, not semi primary.
Of course, any nonfield local domain would work.
It’s local, and since it is a domain, idempotents lift. Its maximal ideal is not nilpotent.