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Turkish Studies - Information Technologies and Applied Sciences, vol.14, no.2, pp.167-177, 2019 (Peer-Reviewed Journal) identifier


In this study, generalized δ-supplementing and amplegeneralized δ-supplementing modules are defined as a newgeneralization of Rad-supplementing modules and modules with theproperty (δ-E) (briefly δ- supplementing) which were studied in(Özdemir, 2016) and (Sözen et. al., 2017) respectively and some basicproperties of them are investigated.While preparing this paper besides the article (Özdemir, 2016), thearticles (Çalışıcı et al., 2012 and Türkmen, 2013) related with moduleshaving a supplement and Rad-supplement in every cofinite extensionare used. Generalized δ-supplementing modules are of course ageneralization of injective modules such as Zöschinger’s modules withthe property ( ) E . It is shown that δ-supplementing and δ- radicalmodules are both generalized δ- supplementing. In general, generalizedδ-supplementing modules need not be δ-supplementing. We give anexample which supports this reality. It is proved that every directsummand of a generalized δ-supplementing module is generalizedδ- supplementing. Moreover, it is proved that generalized δ-supplemented modules are preserved under extensions below somespecial conditions. Following this, it is given that an immediateconsequence such that every module with composition series isgeneralized δ- supplementing. Moreover, it is pointed that the cases being generalized δ-supplementing and injectivity are coincide formodules over δ-V- rings. It is wondered whether the factor module ofa generalized δ- supplementing module is also generalized δ-supplementing and so it is found a positive answer under a specialcondition. It is also proved that for a module M , the necessary and sufficient condition of being ample generalized δ-supplementing is that every submodule ofM is generalized δ-supplementing. As a result, it is obtained that every ample generalized δ-supplementingmodule is a generalized δ- supplemented module.