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Goldie extending modules. (English) Zbl 1214.16005
Commun. Algebra 37, No. 2, 663-683 (2009); Corrigendum 38, No. 12, 4747-4748 (2010); Corrigendum 41, No. 5, 2005 (2013).
In this paper the classical notion of extending module is extended to a notion of \(\mathcal G\)-extending. Roughly speaking \(\mathcal G\)-extending means “extending with respect to the equivalence relation \(\beta\)”, where two submodules \(X\) and \(Y\) are equivalent modulo \(\beta\) if and only if \(X\cap Y\) is essential in both \(X\) and \(Y\). Therefore, \(M\) is \(\mathcal G\)-extending if and only if for each \(X\leq M\) there exists a direct summand \(D\) of \(M\) such that \(X\cap D\) is essential in both \(X\) and \(D\). Basic notions and results are presented in Section 1.
In order to study \(\mathcal G\)-extending modules the authors introduce a new concept of relative ejectivity, which generalizes the concept injectivity with respect a class of exact sequences (Section 2). Then the authors consider the decomposition theory for \(\mathcal G\)-extending modules and give a characterization of the Abelian groups which are \(\mathcal G\)-extending (Section 3). The last part of the paper is dedicated to the study of direct sums of uniform modules (Section 4) and essential extensions which are \(\mathcal G\)-extending (Section 5).

MSC:
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16D50 Injective modules, self-injective associative rings
16D40 Free, projective, and flat modules and ideals in associative algebras
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