On the lattice of strong radicals
In this paper the term ``radical'' means both ``radical class'' and ``radical property''. A radical \(\alpha\) is left (right) strong if every left (right) \(\alpha\)-ideal of any ring \(\mathcal R\) belongs to \(\alpha({\mathcal R})\). A radical \(\alpha\) is said to be strong if it is both left and right strong. It is well-known that the class of all radicals forms a complete lattice with respect to set inclusion. Various authors, including Sands and Puczyłowski, have studied different aspects of this lattice and some other lattices of radicals. The author of this paper answers three open questions of Sands and Puczyłowski, namely: (i) Is the class of all strong radicals a sublattice of the lattice of all radicals? (ii) Is the class of all left strong radicals containing the prime radical a sublattice of the lattice of all radicals? (iii) If \(\alpha\) is a right strong radical containing \(\beta\) and \(J\) is a right ideal of an \(\alpha\)-semisimple ring \(\mathcal R\), does \(\alpha(J)=\beta(J)\) hold? The results of this paper give negative answers to all the above questions.
scientific article; zbMATH DE number 869781