Herhangi bir $S$ halkası ve bir $W$ $S$-modülü için, $W$ modülünün bir $G$ alt modülü, eğer $W/G$ bölüm modülü $\delta$-eşatom ise \emph{eş$_\delta$-eşatom} olarak adlandırılır. Bu çalışmada, ($\oplus$-)\emph{eş$_\delta$-eşatom $\delta$-tümlenmiş modül}, veya kısaca ($\oplus$-)\emph{eş$_\delta$-$\delta$-tümlenmiş modül} terimini her eş$_\delta$-eşatom alt modülü (direkt toplam terimi olan) bir $\delta$-tümleyene sahip olan bir $W$ modülünü belirtmek için tanıtıyoruz. Ayrıca, $W$ modülü, eğer her bir $\delta$-eşatom bölüm modülü, projektif bir $\delta$-örtüye sahipse \emph{eş$_\delta$-eşatom $\delta$-yarı mükemmel} veya kısaca \emph{eş$_\delta$-$\delta$-yarı mükemmel} olarak tanımlanır. Bir $\delta$-yarı mükemmel $S$ halkası üzerinde, $_{S}S$ modülünün $\oplus_{\delta}$-eş-eşatom tümlenmiş olmasının $_{S}S$ modülünün eş$_\delta$-$\delta$-yarı mükemmel olmasına ve $_{S}S$ modülünün $\oplus$-eş$_\delta$-$\delta$-tümlenmiş olmasına denk olduğu kanıtlanmıştır.
Wisbauer R. (1991). Foundations of Modules and Rings. Gordon and Breach Science Publishers, Philadelphia.
Zöschinger H., & Rosenberg F. A. (1980). Koatomare moduln. Mathematische Zeitschrift, 170, 221–232. https://doi.org/10.1007/BF01214862
Alizade R., & Güngör S. (2017). Co-coatomically supplemented modules. Ukrainian Mathematical Journal, 69(7), 1007–1018. https://doi.org/10.1007/s11253-017-1411-x
Koşan M. T., & Harmancı A. (2005). Generalizations of coatomic modules. Open Mathematics, 3(2), 273–281. https://doi.org/10.2478/BF02479203
Alizade R., & Güngör S. (2018). ⊕-co-coatomically supplemented and co-coatomically semiperfect modules. Hacettepe Journal of Mathematics and Statistics, 47(6), 1417–1426. https://dergipark.org.tr/en/pub/hujms
Zhou Y. (2000). Generalizations of perfect, semiperfect and semiregular rings. Algebra colloquium, 7(3), 305–318.
Koşan M. T. (2007). δ-Lifting and δ-supplemented modules. Algebra colloquium, 14(1), 53–60. https://doi.org/10.1142/S1005386707000065
Abdioğlu C., & Şahinkaya S. (2015). Some results on δ-semiperfect rings and δ-supplemented modules. Kyungpook Mathematical Journal, 55, 289–300. https://dx.doi.org/10.5666/KMJ.2015.55.2.289
Eryılmaz F, & Öztürk Sözen E. (2023). On a generalization of ⊕-co-coatomically supplemented modules. Honam Mathmatical Journal, 45(1), 146–159. https://doi.org/10.5831/HMJ.2023.45.1.146
Büyükaşık E., & Lomp C. (2010). When δ-semiperfect rings are semiperfect. Turkish Journal of Mathematics, 34(3), 317–324. https://doi.org/10.3906/mat-0810-15
Tribak R. (2012). Finitely generated δ-supplemented modules are amply δ-supplemented. Bulletin of the Australian Mathematical Society, 86, 430–439. https://doi.org/10.1017/S0004972711003406
Zöschinger H. (1974). Komplemente als direkte summanden. Archiv der Mathematik, 25, 241–253. https://doi.org/10.1007/BF01238671
Özcan A. Ç., & Alkan M. (2006). Duo modules. Glasgow Mathematical Journal, 48, 533–545. https://doi.org/10.1017/S0017089506003260
Garcia J. L. (1989). Properties of direct summands of modules. Communications in Algebra, 17, 73–92. https://doi.org/10.1080/00927878908823714
Tuganbaev A. (1999). Distributive Modules and Related Topics. Gordon and Breach Science Publishers.
Mohamed S. H., & Müller B. J. (1990). Continuous and Discrete Modules. Cambridge University Press.
A Study On a New Generalization of $\delta$-Supplemented Modules
For any ring $S$ and an $S$-module $W$, a submodule $G$ of $W$ is termed \emph{co$_\delta$-coatomic} if the quotient module $W/G$ is $\delta$-coatomic. In this study, we introduce the term ($\oplus$-)\emph{co$_\delta$-coatomically $\delta$-supplemented module}, or shortly ($\oplus$-)\emph{co$_\delta$-$\delta$-supplemented module} to describe a module $W$ where each co$_\delta$-coatomic submodule has a $\delta$-supplement (that is a direct summand) in $W$. Furthermore, a module $W$ is identified as \emph{co$_\delta$-coatomically $\delta$-semiperfect}, or shortly \emph{co$_\delta$-$\delta$-semiperfect}, provided each $\delta$-coatomic quotient module of $W$ has a projective $\delta$-cover. It has been proved that over a $\delta$-semiperfect ring $S$, the module $_{S}S$ is $\oplus_{\delta}$-co-coatomically supplemented if and only if $_{S}S$ is co$_\delta$-$\delta$-semiperfect if and only if $_{S}S$ is $\oplus$-co$_\delta$-$\delta$-supplemented.
Wisbauer R. (1991). Foundations of Modules and Rings. Gordon and Breach Science Publishers, Philadelphia.
Zöschinger H., & Rosenberg F. A. (1980). Koatomare moduln. Mathematische Zeitschrift, 170, 221–232. https://doi.org/10.1007/BF01214862
Alizade R., & Güngör S. (2017). Co-coatomically supplemented modules. Ukrainian Mathematical Journal, 69(7), 1007–1018. https://doi.org/10.1007/s11253-017-1411-x
Koşan M. T., & Harmancı A. (2005). Generalizations of coatomic modules. Open Mathematics, 3(2), 273–281. https://doi.org/10.2478/BF02479203
Alizade R., & Güngör S. (2018). ⊕-co-coatomically supplemented and co-coatomically semiperfect modules. Hacettepe Journal of Mathematics and Statistics, 47(6), 1417–1426. https://dergipark.org.tr/en/pub/hujms
Zhou Y. (2000). Generalizations of perfect, semiperfect and semiregular rings. Algebra colloquium, 7(3), 305–318.
Koşan M. T. (2007). δ-Lifting and δ-supplemented modules. Algebra colloquium, 14(1), 53–60. https://doi.org/10.1142/S1005386707000065
Abdioğlu C., & Şahinkaya S. (2015). Some results on δ-semiperfect rings and δ-supplemented modules. Kyungpook Mathematical Journal, 55, 289–300. https://dx.doi.org/10.5666/KMJ.2015.55.2.289
Eryılmaz F, & Öztürk Sözen E. (2023). On a generalization of ⊕-co-coatomically supplemented modules. Honam Mathmatical Journal, 45(1), 146–159. https://doi.org/10.5831/HMJ.2023.45.1.146
Büyükaşık E., & Lomp C. (2010). When δ-semiperfect rings are semiperfect. Turkish Journal of Mathematics, 34(3), 317–324. https://doi.org/10.3906/mat-0810-15
Tribak R. (2012). Finitely generated δ-supplemented modules are amply δ-supplemented. Bulletin of the Australian Mathematical Society, 86, 430–439. https://doi.org/10.1017/S0004972711003406
Zöschinger H. (1974). Komplemente als direkte summanden. Archiv der Mathematik, 25, 241–253. https://doi.org/10.1007/BF01238671
Özcan A. Ç., & Alkan M. (2006). Duo modules. Glasgow Mathematical Journal, 48, 533–545. https://doi.org/10.1017/S0017089506003260
Garcia J. L. (1989). Properties of direct summands of modules. Communications in Algebra, 17, 73–92. https://doi.org/10.1080/00927878908823714
Tuganbaev A. (1999). Distributive Modules and Related Topics. Gordon and Breach Science Publishers.
Mohamed S. H., & Müller B. J. (1990). Continuous and Discrete Modules. Cambridge University Press.
Önal Kır, E. (2024). A Study On a New Generalization of $\delta$-Supplemented Modules. Sinop Üniversitesi Fen Bilimleri Dergisi, 9(1), 114-127. https://doi.org/10.33484/sinopfbd.1411952