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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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