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$\delta$-Tümlenmiş Modüllerin Yeni Bir Genelleştirilişi Üzerine Bir Çalışma

Year 2024, , 114 - 127, 29.06.2024
https://doi.org/10.33484/sinopfbd.1411952

Abstract

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.

References

  • 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

Year 2024, , 114 - 127, 29.06.2024
https://doi.org/10.33484/sinopfbd.1411952

Abstract

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.

References

  • 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.
There are 16 citations in total.

Details

Primary Language English
Subjects Algebra and Number Theory
Journal Section Research Articles
Authors

Emine Önal Kır 0000-0002-3025-3290

Publication Date June 29, 2024
Submission Date December 29, 2023
Acceptance Date April 3, 2024
Published in Issue Year 2024

Cite

APA 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


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