On ss-Lifting Modules In View of Singularity

Year 2023, Volume: 8 Issue: 2, 145 - 155, 28.12.2023

Esra Öztürk Sözen Elif Eryaşar

https://doi.org/10.33484/sinopfbd.1355648

Abstract

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.

Keywords

Semisimple module, δss-supplemented module, δss-lifting module, Left δss-perfect ring

Singülerlik Açısından ss-Lifting Modüller

Abstract

Bu makalede ss-yükseltilebilir modüllerin singüler versiyonu olan δss-yükseltilebilir modülleri tanımlıyoruz. Çalışmanın amacı δss-yükseltilebilir modüllerden daha genel bir cebirsel yapı elde etmektir. Bir W modülü, her S ≤ W alt modülü için, Socδ(Y ) = δ(Y ) ∩ Soc(Y ) olmak üzere, X ≤ S ve S ∩ Y ≤ Socδ(Y ), koşullarını gerçekleyen W = X ⊕ Y ayrışımına sahip ise W’ya δss-yükseltilebilir modül denir. Bu modüllerin temel özelliklerini araştırıyor ve üzerindeki her modülü δss-yükseltilebilir olan bir halka yapısı arıyoruz. Sonunda ise, δss-mükemmel halkalar aracılığı ile (projektif) δss-yükseltilebilir ve (bol) δss-tümlenmiş modüllerin bir takım karakterizasyonlarını veriyoruz.

Keywords

Yarı basit modül, δss-tümlenmi¸s modül, δss-yükseltilebilir modül, Sol δss-mükemmel halka

References

• Zhou, Y. (2000). Generalizations of perfect, semiperfect and semiregular rings. Algebra Colloquium, 7(3), 305–318. https://doi.org/10.1007/s10011-000-0305-9
• Ungor, B., Halicioglu, S., & Harmanci, A. (2014). On a class of δ-supplemented modules. Bulletin of the Malaysian Mathematical Sciences Society, 37(3), 703–717.
• Nişancı Türkmen, B., & Türkmen, E. (2020). δss-supplemented modules and rings. An St Ovidius Constanta, 28(3), 193–216. https://10.2478/auom-2020-0041
• Koşan, M. (2007). δ-lifting and δ-supplemented modules. Algebra Colloquium, 14(1), 53–60. https://10.1142/S1005386707000065
• Oshiro, K. (1983). Semiperfect modules and quasi-semiperfect modules. Osaka Journal of Mathematics, 20, 337–372. https://10.18910/10960
• Türkmen, E. (2019). Z∗-semilocal modules and the proper class RS. Ukrainian Mathematical Journal, 71(3), 455–469.
• Demirci, Y. M., & Türkmen, E. (2022). WSA-supplements and proper classes. Mathematics, 10(16), 2964.
• Durgun, Y. (2021). SA-supplement submodules. Bulletin of the Korean Mathematical Society, 58(1), 147–161.
• Durğun, Y. (2019). Extended s-supplement submodules. Turkish Journal of Mathematics, 43(6), 2833–2841.
• Alizade, R., Büyükasık, E., & Dur˘gun, Y. (2016). Small supplements, weak supplements and proper classes. Hacettepe Journal of Mathematics and Statistics, 45(3), 649–661.
• Abdioğlu, C., & ¸ Sahinkaya, S. (2015). Some results on δ-semiperfect rings and δ-supplemented modules. Kyungpook Mathematical Journal, 55(2), 289–300. https://10.5666/KMJ.2015.55.2.289
• Clark, J., Lomp, C., Vanaja, N., & Wisbauer, R. (2006). Lifting modules: supplements and projectivity in module theory. Birkhauser Verlag-Basel.
• Mohamed, S., & Müller, B. (1990). Continuous and discrete modules. Cambridge University Press.
• Tribak, R. (2012). Finitely generated δ-supplemented modules are amply δ-supplemented. Bulletin of Australian Mathematical Societyl, 86, 430–439. https://10.1017/S0004972711003406
• Wisbauer, R. (1991). Foundations of module and ring theory: A handbook for study and research. Gordon and Breach Science Publishing.
• Kaynar, E., Türkmen, E., & Çalı ¸sıcı, H. (2020). ss-supplemented modules. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 69(1), 473–485. https://10.31801/cfsuasmas.585727
• Eryılmaz, F. (2021). ss-lifting modules and rings. Miskolc Mathematical Notes, 22(26), 655–662. https://10.18514/MMN.2021.3245
• Kaschl, F. (1982). Modules and rings. Academic Press Inc.
• Talebi, Y., & Talaee, B. (2009). On δ-coclosed submodules. Far East Journal of Mathematical Sciences, 35(1), 19–31.
• Nguyen, X., Ko¸san, M., & Zhou, Y. (2018). On δ-semiperfect modules. Communications in Algebra, 46(11), 4965–4977. https://10.1080/00927872.2018.1459650