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 b3≠0 is not an inner RJ-algebra if there exists a nonzero element a such that a2≠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 nilradical N and a finite-dimensional inner BJ-algebra with a nilradical N.

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