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    A RECENT GENERALIZATION OF RAD-SUPPLEMENTING MODULES
    (2019) Eryılmaz, Figen; Öztürk Sözen, Esra
    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.
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    Bounds For Spectral Radius and Energy of PIS Graphs
    (2024) Öztürk Sözen, Esra; Eryaşar, Elif
    Once the spectral radius and energy of a graph structure have been defined, many properties have been studied. The spectral radius and energy of a graph are related to the eigenvalues of the adjacency matrix of the graph. In this paper, we define an adjacency matrix for a prime ideal sum (PIS) graph and then extend the concepts of spectral radius and energy to PIS graphs. Some bound theorems on the energy and spectral radius of PIS graph structures are given. A SageMath code for plotting these graphs is also provided.
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    On ss-Lifting Modules In View of Singularity
    (2023) Öztürk Sözen, Esra; Eryaşar, Elif
    In this essay we describe δss-lifting modules as a singular version of ss-lifting ones. The focus of this study is to get a more general algebraic structure than ss-lifting modules. A module W is entitled δss-lifting if for each S ≤ W, there occurs a decomposition W = X ⊕ Y with X ≤ S and S ∩ Y ≤ Socδ(Y ), where Socδ(Y ) = δ(Y ) ∩ Soc(Y ). We examine the fundamental properties of this form of modules and also investigate a structure of a ring whose modules are all δss-lifting. Finally, we give several characterizations for (projective) δss-lifting modules and (amply) δss-supplemented modules via δss-perfect rings.