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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 ≠ 0 is not an inner RJ-algebra if there exists a nonzero element a such that a 2 ≠ 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. [ABSTRACT FROM AUTHOR]

The aim of this paper is to describe the irreducible modules for the Jordan algebra of 2 × 2 matrices over an algebraically closed field of characteristic different from 2, 3 and 5 in the class of the commutative power-associative algebras. All irreducible non-unital modules, and irreducible unital modules up to dimension three are classified, namely we find seven non-parameterized and five families of parameterized modules of dimension three. For every k ≥ 2 , an irreducible module of dimension 3k is also constructed. [ABSTRACT FROM AUTHOR]

In this paper we study cohomology and deformations of Jacobi-Jordan algebras. We develop their formal deformation theory. In particular, we introduce a method to construct a versal deformation for a given Jacobi-Jordan algebra, which can induce all deformations and is unique on the infinitesimal level. We construct formal 1-parameter deformations of Jacobi-Jordan algebras up to dimension 5 and versal deformations for 3-dimensional Jacobi-Jordan algebras. [ABSTRACT FROM AUTHOR]