Kyungpook Mathematical Journal 2021; 61(2): 239-248
Published online June 30, 2021
Copyright © Kyungpook Mathematical Journal.
Weak FI-extending Modules with ACC or DCC on Essential Submodules
Adnan Tercan, Ramazan Yaşar*
Department of Mathematics, Hacettepe University, Beytepe Campus, Ankara 06532, Turkey
e-mail : email@example.com
Hacettepe-ASO 1.OSB Vocational School, Hacettepe University, 06938 Sincan Ankara, Turkey
e-mail : firstname.lastname@example.org
Received: April 19, 2020; Accepted: December 14, 2020
In this paper we study modules with the WFI+-extending property. We prove that if M satisfies the WFI+-extending, pseudo duo properties and M/(Soc M) has finite uniform dimension then M decompose into a direct sum of a semisimple submodule and a submodule of finite uniform dimension. In particular, if M satisfies the WFI+-extending, pseudo duo properties and ascending chain (respectively, descending chain) condition on essential submodules then M=M1 ⊕ M2 for some semisimple submodule M1 and Noetherian (respectively, Artinian) submodule M2. Moreover, we show that if M is a WFI-extending module with pseudo duo, C2 and essential socle then the quotient ring of its endomorphism ring with Jacobson radical is a (von Neumann) regular ring. We provide several examples which illustrate our results.
Keywords: CS-module, uniform dimension, ascending chain condition on essential submodules, FI-extending, WFI-extending