Gardner, B. J. Radical theory for algebras with a scheme of operators. (English) Zbl 0615.16001 Acta Math. Hung. 48, 95-107 (1986). 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 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 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 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). Reviewer: R.Wiegandt MSC: 16W50 Graded rings and modules (associative rings and algebras) 16Nxx Radicals and radical properties of associative rings 08A05 Structure theory of algebraic structures Keywords:graded algebras; multioperator groups; schemes of operators; Kurosh- Amitsur radical theory; graded modules; graded rings; Morita contexts; normal radicals Citations:Zbl 0117.259; Zbl 0102.024; Zbl 0295.16002 PDFBibTeX XMLCite \textit{B. J. Gardner}, Acta Math. Hung. 48, 95--107 (1986; Zbl 0615.16001) Full Text: DOI References: [1] V. A. Andrunakievich and Yu. M. Ryabukhin,Radicals of algebras and structure theory (in Russian), Nauka (Moscow, 1979). · Zbl 0507.16009 [2] N. Divinsky, A. Suliński and T. Anderson, Simple rings and invariant radicals,Colloq. Soc. Math. János Bolyai (Theory of Radicals, Eger, 1982), to appear. · Zbl 0583.16002 [3] E. G. Emin, Prevarieties, the groupoid of varieties and strict radicals in the category of modules over all rings (in Russian),Izv. Akad. Nauk Armyanskoi SSR, Matematika,14 (1979) 221–232. [4] B. J. Gardner and P. N. Stewart, Reflected radical classes,Acta Math. Acad. Sci. Hungar.,28 (1976), 293–298. · Zbl 0357.18007 [5] P. J. Higgins, Groups with multiple operators,Proc. London Math. Soc.,6 (1956), 366–416. · Zbl 0073.01704 [6] P. J. Higgins, Algebras with a scheme of operators.Math. Nachr.,27 (1963), 115–132. · Zbl 0117.25903 [7] M. Jaegermann, Normal radicals of endomorphism rings of free and projective modules,Fund. Math.,86 (1975), 237–250. · Zbl 0295.16002 [8] M. Jaegermann, Normal radicals,Fund. Math.,95 (1977), 147–155. · Zbl 0354.16002 [9] A. G. Kurosh, Radicals of rings and algebras,Colloq. Soc. Math. János Bolyai (Rings, Modules and Radicals, Keszthely, 1971), 297–314. [10] R. Mlitz, Radicals and semi-simple classes of {\(\Omega\)}-groups,Proc. Edinburgh Math. Soc. 23 (Series II) (1980), 37–41. · Zbl 0414.17003 [11] B. I. Plotkin,Groups of automorphisms of algebraic systems, Wolters-Noordhoff (Groningen, 1972). [12] B. M. Rudyk, Extensions of modules,Trans. Moscow Math. Soc.,21 (1970), 225–262. [13] Yu. M. Ryabukhin, Radicals in {\(\Omega\)}-groups (in Russian) I,Mat. Issled.,3 (1968) vyp. 2, 123–160; IIibid. Mat. Issled.,3 (1968) vyp. 4, 108–135; IIIibid. Mat. Issled. 4 (1969) vyp. 1, 110–131. · Zbl 0248.20039 [14] A. D. Sands, Radicals and Morita contexts,J. Algebra,24 (1973), 335–345. · Zbl 0253.16007 [15] E. G. Ŝul’geifer, On the general theory of radicals in categories,Amer. Math. Soc. Transl.,59 (1966), 150–162. · Zbl 0192.10501 [16] A. Suliński, Radicals of associative 2-graded rings,Bull. Acad. Polon. Sci. Sér. Sci. Math.,29 (1981), 431–434. · Zbl 0477.16002 [17] A. Suliński and J. F. Watters, On the Jacobson radical of associative 2-graded rings,Acta Math. Hungar., to appear. · Zbl 0533.16005 [18] S. M. Vovsi,Triangular products of group representations and their applications Birkhäuser (Boston-Basel-Stuttgart, 1981). · Zbl 0482.20009 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.