IDEALS IN THE MULTIPLICATION ALGEBRA OF A NONASSOCIATIVE K-ALGEBRA

For a non-associative K-algebra, A, and alpha epsilon A there are K-vector space morphisms L(a)lpha : A --> A and R(a)lpha : A --> A given by L(alpha) : chi bar arrow pointing right alpha chi and R(alpha) : alpha bar arrow pointing right chi alpha. The associative subagebra of End(K)(A) generated by L(alpha), R(alpha) and id(A), for all alpha epsilon A, is called the multiplication algebra of A, and is denoted by M(A). The relation of the structure of M(A) to the structure of A has been studied in [1], [2], [4], and [7]. For each ideal I in M(A) there is a corresponding ideal ZA in A. Similarly for each ideal I in A there is a corresponding ideal (I : A) in M(A). Certain algebraic structure is reflected back and forth through these related ideals. Much that is of interest in rings and algebras, for example radicals, can be expressed in terms of the intersection of certain ideals. We will study, in particular, how the mappings I bar arrow pointing right IA, and I bar arrow pointing right (I : A) respect intersections. These results will then be used to study the radical of A as defined by Albert in [1].