There are many similarities between the categories of \(R\)-modules and \(S\)-acts, but these are not to be found in their respective radical theories. For modules, the natural ingredients used for the radical theory (= torsion theory) are submodules, homomorphic images and one-element modules. In the category of \(S\)-acts, there are not such canonical choices and the properties of the theory very much depend on the type of congruences and factors employed. Torsion assignments for \(S\)-acts are, like for modules, the hereditary Hoehnke radicals. But in contrast to the case for rings and modules, they need not be hereditary Kurosh-Amitsur radicals. The main reason being that for \(S\)-acts congruences need not be Rees congruences.
In this paper radical assignments are required to be Rees congruences. Then, following the general theory of attainable radicals as developed by L. Márki, R. Mlitz and R. Wiegandt [Commun. Algebra 16, No. 2, 249-305 (1988; Zbl 0646.08006)], a typical Kurosh-Amitsur radical theory is developed for \(S\)-acts. This includes the relationship between essential closures of semisimple classes and hereditary radicals. But there are interesting deviations from the usual theory for algebraic categories. For example, within this theory there are non-trivial complementary radical and semisimple classes.
Characterizations of torsion and torsionfree classes of \(S\)-acts are given and it is shown that different hereditary torsion assignments can determine the same torsion class.