The structure of left filial algebras over a field. (Q842026)

A ring \(A\) is left filial if from any chain of left ideals \(I\vartriangleleft_l J\vartriangleleft _l A\), it may be concluded that \(I\vartriangleleft_l A\). Here the authors continue their study of such rings and, in particular, the structure of a left filial algebra over a field is determined. As for rings, an algebra \(A\) over a field \(F\) is left filial if and only if for every \(a\in A\), \(A^*a=A^*a^2+Fa\) (here \(A^*\) denotes the canonical unital \(F\)-algebra extension of \(A\)). Following the initial characterization of semiprime and prime left filial algebras, a structure theorem for the general case is obtained. It is also shown that any algebra which is left filial and filial, must be right filial. The final result is to describe the structure of such algebras.