A note on normal radicals and principally left and right strong radicals. (Q1769343)

The author continues his investigations into principally left (right) hereditary radicals and principally left (right) strong radicals started earlier with \textit{R. Wiegandt} [Algebra Colloq. 8, No. 4, 409-418 (2001; Zbl 1007.16014)]. The relationship between principally left hereditary radicals and principally right hereditary radicals have been studied for many years. It is known that for radicals containing the prime radical \(\beta\) and for radicals contained in \(\beta\), the two notions coincide but in general they are different. In one of the main results, sufficient conditions on a principally right hereditary radical are given to ensure that it is principally left hereditary. This leads to a new characterization of normal radicals. Finally, if \(\gamma\) is a principally right strong radical which satisfies \(A\in\gamma\Rightarrow A^0\in\gamma\) and \(A^0\in\gamma\Rightarrow S\in\gamma\) for any subring \(S\) of \(A^0\), then a necessary and sufficient condition is given to ensure that \(\gamma\) is also a principally left strong radical.