Description of finite-dimensional inner Rickart and Baer Jordan algebras

In the present paper we study the Jordan counterparts of Rickart and Baer *- algebras, i.e., inner RJ-algebras and inner BJ-algebras and prove that a nilpotent Jordan algebra which has no square root nilpotent elements is an inner RJ-algebra. Also we explain that a nilpotent Jordan algebra that has no nilpotent elements with a square root b such that b(3) not equal 0 is not an inner RJ-algebra if there exists a nonzero element a such that a(2) not equal 0.As a main result of the paper we give a description of a finite-dimensional inner RJ-algebra A, isomorphic to R+N, with a nilradicalNand a finite-dimensional inner BJ-algebra with a nilradical N.