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Uniqueness of decomposition, factorisations, \(G\)-groups and polynomials. (English) Zbl 1420.16002

Summary: In this article, we present the classical Krull-Schmidt theorem for groups, its statement for modules due to Azumaya, and much more modern variations on the theme, like the so-called weak Krull-Schmidt theorem, which holds for some particular classes of modules. Also, direct product of modules is considered. We present some properties of the category of \(G\)-groups, a category in which R. Remak’s results about the Krull-Schmidt theorem for groups can be better understood [J. Reine Angew. Math. 139, 293–308 (1911; JFM 42.0156.01)]. In the last section, direct-sum decompositions and factorisations in other algebraic structures are considered.

MSC:

16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16D90 Module categories in associative algebras
18E05 Preadditive, additive categories

Citations:

JFM 42.0156.01
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