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Retracted: Upper basic and congruent submodules of QTAG-modules. (English) Zbl 1221.16003

Sci., Ser. A, Math. Sci. (N.S.) 16, 95-103 (2008); retracted ibid. 27, 99-101 (2016).
Summary: The cardinality of the minimal generating set of a module \(M\), i.e. \(g(M)\), plays a very important role in the study of QTAG-Modules. L. Fuchs [Infinite Abelian groups. Vol. I. Academic Press (1970; Zbl 0209.05503)] mentioned the importance of upper and lower basic subgroups of primary groups. A need was felt to generalize these concepts for modules. An upper basic submodule \(B\) of a QTAG-Module \(M\) reveals much more information about the structure of \(M\). We find that each basic submodule of \(M\) is contained in an upper basic submodule and contains a lower basic submodule.
Two submodules \(N,K\subseteq M\) are congruent if there exists an automorphism of \(M\) which maps \(N\) onto \(K\). In this case \(M/N\cong M/K\) and \(N\cong K\), but these conditions are not sufficient for the congruence. This motivates us to find sufficient conditions in terms of Ulm invariants and the extensions of height preserving isomorphisms of submodules.
Editorial remark: According to the retraction notice [Zbl 1453.16005], this paper has been retracted since is largely identical to [P. Hill, Mich. Math. J. 18, 187–192 (1971; Zbl 0223.20057)] and [P. Hill and C. Megibben, Lect. Notes Math. 1006, 513–518 (1983; Zbl 0517.20028)].

MSC:

16D80 Other classes of modules and ideals in associative algebras
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
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