Jordan counterparts of Rickart and Baer ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}-algebras, II

We introduce and investigate new classes of Jordan algebras which are close to but wider than Rickart and Baer Jordan algebras. Such Jordan algebras are called weak RJ- and weak BJ-algebras respectively. Criterions are given for a Jordan algebra to be a weak BJ-algebra. Also, it is proved that every finite dimensional Jordan algebra A, the degenerate radical of which does not have nilpotent elements with a square root in A and the quotient with respect to this radical of which has no nilpotent elements, is a weak BJ-algebra.