Radical theory for algebras with a scheme of operators (Q1820844)

The author defines \(<I,\Omega >\)-graded algebras where \(<I,\Omega >\) is an algebra with a set \(\Omega\) of (possibly partial) finitary operations. His graded algebras are generalizations of multioperator groups (where instead of an underlying set there are several) and special cases of algebras with schemes of operators introduced by \textit{P. J. Higgins} [Math. Nachr. 27, 115-132 (1963; Zbl 0117.259)]. It is proved that varieties of these graded algebras form categories which satisfy the axioms of \textit{E. G. Shul'gejfer} [Mat. Sb., Nov. Ser. 51(93), 487-500 (1960; Zbl 0102.024)] and hence these categories are appropriate for developing a Kurosh-Amitsur radical theory. Examples for such graded algebras are graded modules, graded rings, group homomorphisms, complexes, modules over various rings, group representations and Morita contexts. Generalizing normal radicals of \textit{M. Jaegermann} [Fundam. Math. 86, 237-250 (1975; Zbl 0295.16002)] the author introduces the notion of a normal family of radicals getting in this way radical properties for a multiset structure by radical properties of the components (which are multioperator groups).