Mathematica Bohemica, Vol. 148, No. 2, pp. 211-222, 2023


Eventually semisimple weak $FI$-extending modules

Figen Takıl Mutlu, Adnan Tercan, Ramazan Yaşar

Received July 2, 2021.   Published online June 2, 2022.

Abstract:  In this article, we study modules with the weak $FI$-extending property. We prove that if $M$ satisfies weak $FI$-extending, pseudo duo, $C_3$ properties and $M/{\rm Soc} M$ has finite uniform dimension then $M$ decomposes into a direct sum of a semisimple submodule and a submodule of finite uniform dimension. In particular, if $M$ satisfies the weak $FI$-extending, pseudo duo, $C_3$ properties and ascending (or descending) chain condition on essential submodules then $M=M_1øplus M_2$ for some semisimple submodule $M_1$ and Noetherian (or Artinian, respectively) submodule $M_2$. Moreover, we show that a nonsingular weak $CS$ (or weak $C_{11}^*$, or weak $FI$) module has a direct summand which essentially contains the socle of the module and is a $CS$ (or $C_{11}$, or $FI$-extending, respectively) module.
Keywords:  $CS$-module; weak $CS$-module; uniform dimension; ascending chain on essential submodules; $C_{11}$-module; $FI$-extending; weak $FI$-extending
Classification MSC:  16D50, 16D80


References:
[1] F. W. Anderson, K. R. Fuller: Rings and Categories of Modules. Graduate Texts in Mathematics 13. Springer, New York (1992). DOI 10.1007/978-1-4612-4418-9 | MR 1245487 | Zbl 0765.16001
[2] E. P. Armendariz: Rings with DCC on essential left ideals. Commun. Algebra 8 (1980), 299-308. DOI 10.1080/00927878008822460 | MR 0558116 | Zbl 0444.16015
[3] G. F. Birkenmeier, G. Călugăreanu, L. Fuchs, H. P. Goeters: The fully invariant extending property for abelian groups. Commun. Algebra 29 (2001), 673-685. DOI 10.1081/AGB-100001532 | MR 1841990 | Zbl 0992.20039
[4] G. F. Birkenmeier, B. J. Müller, S. T. Rizvi: Modules in which every fully invariant submodule is essential in a direct summand. Commun. Algebra 30 (2002), 1395-1415. DOI 10.1080/00927870209342387 | MR 1892606 | Zbl 1006.16010
[5] G. F. Birkenmeier, J. K. Park, S. T. Rizvi: Extensions of Rings and Modules. Birkhäuser, New York (2013). DOI 10.1007/978-0-387-92716-9 | MR 3099829 | Zbl 1291.16001
[6] V. Camillo, M. F. Yousif: CS-modules with ACC or DCC on essential submodules. Commun. Algebra 19 (1991), 655-662. DOI 10.1080/00927879108824160 | MR 1100368 | Zbl 0718.16006
[7] N. V. Dung, D. V. Huynh, P. F. Smith, R. Wisbauer: Extending Modules. Pitman Research Notes in Mathematics Series 313. Longman, Harlow (1994). DOI 10.1201/9780203756331  | MR 1312366 | Zbl 0841.16001
[8] K. R. Goodearl: Singular torsion and the splitting properties. Mem. Am. Math. Soc. 124 (1972), 89 pages. DOI 10.1090/memo/0124 | MR 0340335 | Zbl 0242.16018
[9] K. R. Goodearl: Ring Theory: Nonsingular Rings and Modules. Pure and Applied Mathematics, Marcel Dekker 33. Marcel Dekker, New York (1976). MR 0429962 | Zbl 0336.16001
[10] I. Kaplansky: Infinite Abelian Groups. University of Michigan Press, Ann Arbor (1969). MR 0233887 | Zbl 0194.04402
[11] P. F. Smith: CS-modules and weak CS-modules. Non-Commutative Ring Theory. Lecture Notes in Mathematics 1448. Springer, Berlin (1990), 99-115. DOI 10.1007/BFb0091255 | MR 1084626 | Zbl 0714.16007
[12] P. F. Smith: Modules with many direct summands. Osaka J. Math. 27 (1990), 253-264. MR 1066623 | Zbl 0703.16007
[13] P. F. Smith, A. Tercan: Generalizations of CS-modules. Commun. Algebra 21 (1993), 1809-1847. DOI 10.1080/00927879308824655 | MR 1215548 | Zbl 0779.16002
[14] P. F. Smith, A. Tercan: Direct summands of modules which satisfy $(C_{11})$. Algebra Colloq. 11 (2004), 231-237. MR 2058772 | Zbl 1075.16002
[15] A. Tercan: On certain CS-rings. Commun. Algebra 23 (1995), 405-419. DOI 10.1080/00927879508825228 | MR 1311796 | Zbl 0820.16001
[16] A. Tercan: Weak $(C_{11}^+)$ modules with ACC or DCC on essential submodules. Taiwanese J. Math. 5 (2001), 731-738. DOI 10.11650/twjm/1500574991 | MR 1870043 | Zbl 1015.16001
[17] A. Tercan: Eventually weak $(C_{11})$ modules and matrix $(C_{11})$ rings. Southeast Asian Bull. Math. 27 (2003), 729-737. MR 2045380 | Zbl 1063.16006
[18] A. Tercan, R. Yaşar: Weak $FI$-extending modules with ACC or DCC on essential submodules. Kyungpook J. Math. 61 (2021), 239-248. DOI 10.5666/KMJ.2021.61.2.239 | MR 4284189 | Zbl 07445263
[19] A. Tercan, C. C. Yücel: Module Theory, Extending Modules and Generalizations. Frontiers in Mathematics. Birkhäuser, Basel (2016). DOI 10.1007/978-3-0348-0952-8 | MR 3468915 | Zbl 1368.16002
[20] R. Yaşar: Modules in which semisimple fully invariant submodules are essential in summands. Turk. J. Math. 43 (2019), 2327-2336. DOI 10.3906/mat-1906-36 | MR 4020392 | Zbl 1431.16006

Affiliations:   Figen Takil Mutlu, Eskişehir Technical University, Department of Mathematics, 26555 Eskisehir, Turkey, e-mail: figent@eskisehir.edu.tr; Adnan Tercan, Hacettepe University, Department of Mathematics, 06800 Beytepe/Ankara, Turkey, e-mail: tercan@hacettepe.edu.tr; Ramazan Yaşar, Hacettepe University, Hacettepe-ASO 1.OSB Vocational School, Türkmenistan Cd. ASORA İş Merkezi, 06938 Sincan/Ankara, Turkey, e-mail: ryasar@hacettepe.edu.tr


 
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