Erratum to: On base radical and semisimple classes defined by class operators

Abstract

Regarding the article of the title, which appeared in Acta Math. Hungar., 138 (2013), 307–328, we regret to note that some of the arguments given in Section 6 are faulty. In particular, the proof of Theorem 6.3 is flawed (even if one corrects its statement to the more plausible claim that for every $${a \in A}$$ $${\mathcal{X}_b(A) \in S(\mathcal{R}))}$$ . Hence also Corollaries 6.4 and 6.5 as well as Proposition 6.7 are in doubt as stated. However, the main claim of the section, that every base radical/semisimple pair arises from a Hoehnke radical is still correct, although a different proof is required. In particular the stated definition of $${\mathcal{X}_b}$$ must be altered in order that the arguments work. (Perhaps the earlier definition can be made to work, but we do not see how.) We repeat the general definition of a Hoehnke radical operation on a universal class below, and follow it with the needed new arguments.