On the lattice of matric-extensible radicals.
The paper deals with the class \(\mathbb{L}_{mh}\) of hereditary matric-extensible radicals. It is known from \textit{R. L. Snider} [Pac. J. Math. 40, 207-220 (1972; Zbl 0235.16006)] that the class \(\mathbb{L}\) of radicals of rings has a complete lattice structure, and the classes \(\mathbb{L}_h\) of hereditary radicals, and \(\mathbb{L}_m\) of matric-extensible radicals are both complete sublattices in \(\mathbb{L}\). It is proved that if \(S\) is a simple ring then the lower radical \({\mathcal L}\{S\}\) determined by \(\{S\}\) is an atom in \(\mathbb{L}_{mh}\) if and only if (a) \(S^2=0\) or (b) \(S\) is idempotent and both \(M_n\)-unique and \(M_n\)-invariant for all \(n\in\mathbb{N}\). The main result of the paper states that \(\mathbb{L}_{mh}\) is atomic and the atoms are exactly the radicals \({\mathcal L}_{mh}\{S\}=\bigwedge\{\rho\mid\rho\in\mathbb{L}_{mh}\), \(S\in\rho\}\) with \(S\) a simple ring. It is worth mentioning that the paper presents nice and interesting examples.
scientific article; zbMATH DE number 2099084