On the upper radical determined by filial rings. (Q858100)

A ring \(R\) (associative but not necessarily with unity) is called filial, if \(A\trianglelefteq R\) and \(B\trianglelefteq A\) implies \(B\trianglelefteq R\). Left filial rings are defined similarly using left ideals. The class of filial rings determines an upper radical \(\mathcal X\), which was studied by \textit{G. Tzintzis} [Acta Math. Hung. 49, 173-184 (1987; Zbl 0611.16005) and ibid. 49, 307-314 (1987; Zbl 0631.16004)]. In this paper, the authors extend and simplify several known results, give a description of \(\mathcal X\), and answer in the negative some open problems posed by Tzintzis.