A comparison of Kurosh-Amitsur and base radical classes (Q2051356)

The authors continue their investigations into the theory of base radicals and its similarities and differences with Kurosh-Amitsur radicals. The setting is a universal class of objects \(\mathcal{A}\), called \(\mathcal{P }\)-algebras, which fulfills suitable conditions to make the development of a radical theory viable as defined by \textit{E. R. Puczyłowski} [Algebra Univers. 30, No. 1, 53--60 (1993; Zbl 0784.08009)]. Their approach depends on the class operators \(\mathbf{U}\), \(\mathbf{S}\) and \(\mathbf{S}^{\prime }\) acting on subclasses \(\mathcal{X}\) of \(\mathcal{A}\) defined by \(\mathbf{U}(\mathcal{X})=\{A\in \mathcal{A}|A\) has no nonzero factor in \(\mathcal{X\}}\), \(\mathbf{S}(\mathcal{X})=\{A\in \mathcal{A}|A\) has no nonzero accessible subalgebra in \(\mathcal{X}\}\), and \(\mathbf{S} ^{\prime }(\mathcal{X})=\{A\in \mathcal{A}|A\) has no nonzero ideal in \( \mathcal{X}\}\). By definition, for any subclass \(\mathcal{X}\subseteq \mathbf{A}\), \(\mathcal{X}\) is a \textit{K--A radical class} whenever \( \mathcal{X}=\mathbf{US}^{\prime }(\mathcal{X});\) a \textit{base radical class } whenever \(\mathcal{X}=\mathbf{US}(\mathcal{X});\) a \textit{K-A semisimple class} whenever \(\mathcal{X}=\mathbf{S}^{\prime }\mathbf{U}(\mathcal{X});\) and a \textit{base semisimple class} whenever \(\mathcal{X}=\mathbf{SU}( \mathcal{X})\). In general, the relation of being an ideal is not a transitive relation. For this reason, the notion of an ideal is often replaced with being an accessible subobject. In universal classes where semisimple classes are hereditary, as for example for associative rings, the notions KA-radical and base radical coincide. The theory of base radicals is thus a useful tool to investigate the consequences of replacing ideal with accessible subobject in radical theory of universal classes where semisimple classes need not be hereditary. It is shown, amongst others, that the largest hereditary subclass of a base radical class need not be a base radical. This is in contrast to the result of Puczyłowski that the largest hereditary subclass of a K-A radical class is a K-A radical class [\textit{E. R. Puczylowski}, Acta Sci. Math. 44, 133--135 (1982; Zbl 0488.16008)]. The existence of an interesting class of algebras follows from their theory, namely a class of algebras which is not a K-A radical-semisimple class, nor a base radical-semisimple class but rather a K-A radical-base semisimple class.