I am searching for a ring $R$ (preferably semisimple) and a nonzero primitive idempotent $e\in R$ such that the corner ring $eRe$ is not local. I am inspired by the theorems on semiperfect rings in which there are idempotents $e_i$ with each corner ring $e_iRe_i$ a local ring. I have also scanned the ERT site (the encyclopedia of ring theory), but to no avail. Thanks for any cooperation!
2026-03-25 15:54:28.1774454068
A non-local corner ring
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You seek a primitive idempotent which is not a local idempotent.
This is not possible in a semisimple ring, since these two notions coincide for semiperfect rings (Proposition 23.5 in Lam's First course in noncommutative rings)
Without additional conditions, it is fairly easy to find such idempotents: for example, $1\in \mathbb Z$ is primitive but not local.