Rings whose nilpotent elements form a Levitzki radical ring

It is shown that a locally 2-primal ring, hut not 2-primal, can be always constructed from any given a 2-primal ring. Locally 2-primal rings are NI but we show that there are NI rings which are not locally 2-primal. We prove that every minimal noncommutative (locally) 2-primal ring is isomorphic to the 2 by 2 upper triangular matrix ring over GF(2). By Smoktunowicz (2000), a nit ring R need not be locally 2-primal, but we show that it is the case if and only if R is a Levitzki radical ring. We also prove that the local 2-primalness is inherited by polynomial rings, but not by power series rings. However in the case of rings of hounded index of nilpotency, a ring is locally 2-primal if and only if so is its power series ring.