Burcu NİŞANCI TÜRKMEN

WEAKLY LOCALLY ARTINIAN SUPPLEMENTED MODULES

In this study, by using the concept of locally artinian supplemented modules, we have obtained that weakly locally artinian supplemented modules as a proper generalization of these modules in module theory. Our results generalize and extend various comparable results in the existing literature. We have proved that a notion of weakly locally artinian supplemented modules inherited by factor modules, finite sums and small covers. We have obtained that weakly locally artinian supplemented modules with small radical coincide with weakly (radical) supplemented modules which have locally artinian radical. Also, we have shown that if $N$ and $\frac{M}{N}$ are weakly locally artinian supplemented for some submodule $N\subseteq M$ which has a weak locally artinian supplement in $M$ then $M$ is weakly locally artinian supplemented.

Keywords:

Locally artinian module, (weak) supplement, (weak) locally artinian supplement (weakly) locally artinian supplemented module,

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R. Alizade, E. Büyükaşık. Extensions of weakly supplemented modules, Math. Scand., Vol. 103, pp. 161-168 (2008).
E. Büyükaşık, E. Türkmen. Strongly radical supplemented modules. Ukr. Math. J., 63, No. 8, 1306-1313 (2011).
E. Kaynar, H. Çalışıcı, E. Türkmen. ss-supplemented modules. Communications Faculty of Science University of Ankara Series A1 Mathematics and statistics, Vol. 69, 1, pp 473-485 (2020). D. Keskin. On lifting modules, Comm. Algebra, Vol.28:7, 3427-3440 (2000).
T.Y. Lam. A first course in noncommutative rings, Springer, New York (1999).
C. Lomp. On semilocal modules and rings, Comm. Algebra, Vol. 27:4, 1921-1935 (1999).
B. Nişancı Türkmen, E. Türkmen. On a generalization of weakly supplemented modules. An. Stiin. Univ. Al. I. Cuza Din Iasi. Math (N.S.), Vol. 63(2), pp. 441-448 (2017).
D. W. Sharpe, P. Vamos. Injective modules. Cambridge University Press, Cambridge, (1972).
Y. Şahin, B. Nişancı Türkmen. Locally-artinian supplemented modules. 9th International Eurasian Conference On Mathematical Sciences and Applications Abstract Book,Skopje, North Macedonia, pp. 26 (2020).
D. Van Huynh, R. Wisbauer. Characterization of locally artinian modules. Journal of Algebra, Vol. 132, pp. 287-293 (1990).
R. Wisbauer. Foundations of modules and rings. Gordon and Breach, Springer-Verlag (1991).
D. X. Zhou, X. R. Zhang. Small-essential submodules and morita duality. Southeast Asian Bulletin of Mathematics, Vol. 35, pp 1051-1062 (2011).
H. Zöschinger. Moduln, die in jeder erweiterung ein komplement haben. Math. Scand., Vol. 35, pp. 267-287 (1974).
H. Zöschinger. Komplementierte modulnüber dedekindringen. Journal of Algebra, Vol. 29, pp. 42-56 (1974).