Jordan counterparts of Rickart and Baer $*$-algebras
  There are Jordan analogues of annihilators in Jordan algebras which are called Jordan annihilators. The present paper is devoted to investigation of those Jordan algebras every Jordan annihilator of which is generated by an idempotent as an inner ideal. We prove that a finite dimensional unital Jordan algebra satisfies this condition if and only if it has no nilpotent elements, and in this case it is a direct sum of simple Jordan algebras. 
Shavkat Ayupov and Farhodjon Arzikulov
math.OA math.RA