Her Eş-sonlu Genişlemesinde Zayıf Rad-tümleyene Sahip Modüller

Yıl 2024, Cilt: 27 Sayı: 1, 379 - 385, 29.02.2024

Emine Önal Kır , Hamza Çalışıcı

https://doi.org/10.2339/politeknik.991518

Öz

Bu çalışmada, (CWE) ve (CWEE) özelliğine sahip modüllerin bir genelleştirilişi olarak (CWRE) ve (CWREE) özelliğine sahip modülleri tanımlıyoruz. R bir halka ve M sol R-modül olsun. Eğer M (CWRE) özelliğine sahip ise, M modülünün her direkt toplam terimi (CWRE) özelliğine sahiptir. Bir R halkasının yarıyerel olması için gerek ve yeter şart her sol R-modülün (CWRE) özelliğine sahip olmasıdır. Ayrıca (WRE*) ve (WREE*) özelliğine sahip modülleri çalışıyoruz. Bir modülün (WREE*) özelliğine sahip olması için gerek ve yeter şart onun her alt modülünün (WRE*) özelliğine sahip olmasıdır. 

Anahtar Kelimeler

Zayıf Rad-tümleyen, yarıyerel halka, eş-sonlu genişleme

Modules Having a Weak Rad-Supplement in Every Cofinite Extension

Öz

In this study, we introduce the modules with the properties (CWRE) and (CWREE) as a generalization of the modules with the properties (CWE) and (CWEE). Let R be a ring and M be a left R-module. If M has the property (CWRE), then every direct summand of M has the property (CWRE). A ring R is semilocal if and only if every left R-module has the property (CWRE). We also study the modules that have the properties (WRE*) and (WREE*). A module has the property (WREE*) if and only if every submodule of it has the property (WRE*).   

Anahtar Kelimeler

Weak Rad-supplement, semilocal ring, cofinite extension

Kaynakça

• [1] Dalkılıç O. and Demirtaş N., “VFP-soft kümeler ve karar verme problemleri üzerine uygulaması”, Politeknik Dergisi, 24(4):1391-1399, (2021).
• [2] Güler E., “Rotational hypersurfaces satisfying〖 ∆〗^I R=AR in the four-dimensional Euclidean space”, Politeknik Dergisi, 24(2):517-520, (2021).
• [3] Karadağ M., “A note on nearly hyperbolic cosymplectic manifolds”, Politeknik Dergisi, 23(4):1403-1406, (2020).
• [4] Sharpe D. W. and Vamos P., “Injective Modules”, Lecturers in Pure Mathematics University of Sheffield, Cambridge at the University Press, (1972).
• [5] Alizade R., Bilhan G. and Smith P. F., “Modules whose maximal submodules have supplements”, Communications in Algebra, 29 (6): 2389-2405, (2001).
• [6] Çalışıcı H. and Türkmen E., “Modules that have a supplement in every cofinite extension”, Georgian Mathematical Journal, 19: 209-216 (2012).
• [7] Zöschinger H., “Komplementierte moduln über Dedekindringen”, Journal of Algebra, 29: 42-56 (1974).
• [8] Wisbauer R., “Foundations of modules and rings”, Gordon and Breach, Philadelphia, (1991).
• [9] Wang Y. and Ding N., “Generalized supplemented modules”, Taiwanese Journal of Mathematics, 10(6): 1589-1601 (2006).
• [10] Clark J., Lomp C., Vanaja N. and Wisbauer R., “Lifting modules. Supplements and Projectivity in Module Theory”, Frontiers in Mathematics, Birkhäuser, Basel, (2006).
• [11] Lomp C., “ On semilocal modules and rings”, Communications in Algebra, 27 (4): 1921-1935 (1999).
• [12] Zöschinger H., “Invarianten wesentlicher Überdeckungen”, Mathematische Annalen, 237: 193-202 (1978).
• [13] Choubey S. K., Pandeya B. M. and Gupta A. J., “Amply weak Rad-supplemented modules”, International Journal of Algebra, 6 (27): 1335-1341, (2012).
• [14] Zöschinger H., “Moduln, die in jeder erweiterung ein komplement haben”, Mathematica Scandinavica, 35: 267-287 (1974).
• [15] Polat N. M., Çalışıcı H. and Önal E., “ Modules that have a weak supplement in every cofinite extension”, Palestine Journal of Mathematics, 4(1): 553-556 (2015).
• [16] Nişancı Türkmen B., “Modules that have a supplement in every coatomic extension”, Miskolc Mathematical Notes, 16 (1): 543-551 (2015).
• [17] Önal E., Çalışıcı H. and Türkmen E., “ Modules that have a weak supplement in every extension”, Miskolc Mathematical Notes, 17(1): 471-481, (2016).
• [18] Eryılmaz F. Y. and Eren Ş., “Totally cofinitely weak Rad-supplemented modules”, International Journal of Pure and Applied Mathematics, 80 (5): 683-692 (2012).
• [19] Büyükaşık E. and Lomp C., “On a recent generalization of semiperfect rings”, Bulletin of the Australian Mathematical Society, 78: 317-325, (2008).
• [20] Puninski G., “Some model theory over a nearly simple uniserial domain and decompositions of serial modules”, Journal of Pure and Applied Algebra , 163 (3): 319-337 (2001).
• [21] Nişancı Türkmen B., “Modules that have a Rad- supplement in every cofinite extension”, Miskolc Mathematical Notes, 14 (3): 1059-1066 (2013).
• [22] Alizade R. and Büyükaşık E., “Cofinitely weak supplemented modules”, Communications in Algebra, 31 (11):5377-5390, (2003).
• [23] Mohamed S. H. and Müller B. J., “Continous and discrete modules”, London Mathematical Society Lecture Note Series 147, Cambridge University Press, (1990).
• [24] Önal Kır E. and Çalışıcı H., “Modules that have a weak Rad-supplement in every extension”, Journal of Science and Arts, 3 (44): 611-616 (2018).