On left stable radical classes (Q2718040)

A radical class is left stable if its semisimple class is left hereditary. A class \(M\) of (semi)prime rings is a left stable (weakly) special class, if it is closed under essential extensions and satisfies condition (*): \(0\neq J\vartriangleleft L\vartriangleleft_\ell R\in M\) implies \(0\neq J/r(J,J)\in M\) where \(r(J, J)\) is the right annihilator of \(J\). It is proved that a supernilpotent radical is left stable if and only if its semisimple class is a left stable weakly special class, the upper radical of a left stable weakly special class is a left stable supernilpotent radical; corresponding results hold for special radicals and left stable special classes. If \(M\) is a class of (semi)prime rings satisfying condition (*), then its essential cover \(M_k\) is the smallest left stable (weakly) special class containing \(M\).