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Weak F I-extending Modules with ACC or DCC on Essential Submodules
Tercan, Adnan,Yasar, Ramazan Department of Mathematics 2021 Kyungpook mathematical journal Vol.61 No.2
In this paper we study modules with the W F I<sup>+</sup>-extending property. We prove that if M satisfies the W F I<sup>+</sup>-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 W F I<sup>+</sup>-extending, pseudo duo properties and ascending chain (respectively, descending chain) condition on essential submodules then M = M<sub>1</sub> ⊕ M<sub>2</sub> for some semisimple submodule M<sub>1</sub> and Noetherian (respectively, Artinian) submodule M<sub>2</sub>. Moreover, we show that if M is a W F I-extending module with pseudo duo, C<sub>2</sub> 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.
Goldie extending property on the class of $z$-closed submodules
Adnan Tercan,Ramazan Yasar,Canan Celep Yucel 대한수학회 2022 대한수학회보 Vol.59 No.2
In this article, we define a module $M$ to be $G^{\, z}$-extending if and only if for each $z$-closed submodule $X$ of $M$ there exists a direct summand $D$ of $M$ such that $X\cap D$ is essential in both $X$ and $D$. We investigate structural properties of $G^{\, z}$-extending modules and locate the implications between the other extending properties. We deal with decomposition theory as well as ring and module extensions for $G^{\, z}$-extending modules. We obtain that if a ring is right $G^{\, z}$-extending, then so is its essential overring. Also it is shown that the $G^{\, z}$-extending property is inherited by its rational hull. Furthermore it is provided some applications including matrix rings over a right $G^{\, z}$-extending ring.
The π-extending Property via Singular Quotient Submodules
Kara, Yeliz,Tercan, Adnan Department of Mathematics 2019 Kyungpook mathematical journal Vol.59 No.3
A module is said to be ${\pi}$-extending provided that every projection invariant submodule is essential in a direct summand of the module. In this article, we focus on the class of modules having the ${\pi}$-extending property by looking at the singularity of quotient submodules. By doing so, we provide counterexamples, using hypersurfaces in projective spaces over complex numbers, to show that being generalized ${\pi}$-extending is not inherited by direct summands. Moreover, it is shown that the direct sums of generalized ${\pi}$-extending modules are generalized ${\pi}$-extending.
When Some Complement of an EC-Submodule is a Direct Summand
Denizli, Canan Celep Yucel,Ankara, Adnan Tercan Department of Mathematics 2010 Kyungpook mathematical journal Vol.50 No.1
A module M is said to satisfy the $EC_{11}$ condition if every ec-submodule of M has a complement which is a direct summand. We show that for a multiplication module over a commutative ring the $EC_{11}$ and P-extending conditions are equivalent. It is shown that the $EC_{11}$ property is not inherited by direct summands. Moreover, we prove that if M is an $EC_{11}$-module where SocM is an ec-submodule, then it is a direct sum of a module with essential socle and a module with zero socle. An example is given to show that the reverse of the last result does not hold.