Abstract
In the early sixties Andrunakievič and Rjabuhin extended the general theory of radicals for rings and groups to modules over associative rings ([1],[2]). As in the case of rings and groups in the work of Kuraš and Amitsur, the modules have to satisfy some axiomatic conditions in order to define an appropriate concept of radical, a so-called general class of modules. In this note we use their concepts of general class of modules with the corresponding radical class and develop it further. For any radical ℝ there exists a general class of modules ∑, such that the radical class corresponding to ∑, coincides with ℝ (Theorem 1). A radical ℝ (in the class of associative rings) is hereditary if for every ring A and any ideal B of A the equality ℝ(B) = B ∩ ℝ(A) holds. Analogous results hold for the ∑-radical of a class ∑ of modules (Propositions 8,9 and Theorem 2). For unexplained notions we refer to [1].
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Bibliography
V.A. ANDRUNAKIEVIČ AND JU.M. RJABUHIN Modules and radicals, Dokl. Akad. Nauk. SSSR 156(1964), 991–994. (Russian)
V.A. ANDRUNAKIEVIČ AND JU.M. RJABUHIN Special modules and special radicals, Dokl. Akad.Nauk. SSSR 147(1962), 1274–1277 (Russian).
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© 1983 Springer-Verlag Berlin Heidelberg
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van Leeuwen, L.C.A. (1983). On Modules and Radicals. In: Göbel, R., Lady, L., Mader, A. (eds) Abelian Group Theory. Lecture Notes in Mathematics, vol 1006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21560-9_46
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DOI: https://doi.org/10.1007/978-3-662-21560-9_46
Publisher Name: Springer, Berlin, Heidelberg
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