Facchini, Alberto; Şahinkaya, Serap Uniqueness of decomposition, factorisations, \(G\)-groups and polynomials. (English) Zbl 1420.16002 Int. Electron. J. Algebra 24, 107-128 (2018). 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. Cited in 1 Document 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 Keywords:Krull-Schmidt theorem; indecomposable module; direct-sum decomposition; endomorphism ring; uniserial module; Artinian module Citations:JFM 42.0156.01 PDFBibTeX XMLCite \textit{A. Facchini} and \textit{S. Şahinkaya}, Int. Electron. J. Algebra 24, 107--128 (2018; Zbl 1420.16002) Full Text: DOI References: 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.