Article
Kyungpook Mathematical Journal 2021; 61(2): 239248
Published online June 30, 2021
Copyright © Kyungpook Mathematical Journal.
Weak FIextending Modules with ACC or DCC on Essential Submodules
Adnan Tercan, Ramazan Yaşar^{*}
Department of Mathematics, Hacettepe University, Beytepe Campus, Ankara 06532, Turkey
email : tercan@hacettepe.edu.tr
HacettepeASO 1.OSB Vocational School, Hacettepe University, 06938 Sincan Ankara, Turkey
email : ryasar@hacettepe.edu.tr
Received: April 19, 2020; Accepted: December 14, 2020
Abstract
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=M_{1} ⊕ M_{2} for some semisimple submodule M_{1} and Noetherian (respectively, Artinian) submodule M_{2}. Moreover, we show that if M is a WFIextending module with pseudo duo, C_{2} 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: CSmodule, uniform dimension, ascending chain condition on essential submodules, FIextending, WFIextending
1. Introduction
Assume that all rings are associative and have identity elements and all modules are unital right modules. Let
Armendariz [2, Proposition 1.1] proved that a module

(i)
M/N has finite uniform dimension for every essential submoduleN ofM ,

(ii) every homomorphic image of
$M/(\text{Soc}M)$ has finite uniform dimension.
Camillo and Yousif [6, Corollary 3] proved that if
A module
A module
A module
No other implications can be added to this table in general. To see why this is the case, please consult [19]. Note that it is an open problem to determine whether the
The purpose of this paper is to try to extend the result of [16, Theorem 11, Corollary 12] to
2. Weak FI^{+} extending Modules
Let
Example 2.1.
Let
For more examples similar to Example 2.1 (see [17, Corollary 16]). Surprisingly, Example 2.1 and [17, Corollary 16] also show that we can not replace
Theorem 2.2.
Let
Now
The next example shows that
Example 2.3.
Let Kbe a field and Van infinite dimensional vector space over K. Let Rbe the trivial extension Kwith Vi.e. ,
Then
Corollary 2.4.
Let
Now, let us think of general modules over arbitrary rings. Since we require both the pseudo duo property and that
Example 2.5.

(i) Let
M be the free$\mathbb{Z}$ module of infinite rank i.e.,$\mathbb{Z}M={\oplus}_{i=1}^{\infty}\mathbb{Z}$ . Then$\text{Soc}{M}_{\mathbb{Z}}=0$ . HenceM satisfies pseudo duo property. However,$M/(\text{Soc}M)\cong {M}_{\mathbb{Z}}$ which has infinite uniform dimension.

(ii) Let
R be a prime ring and let${M}_{R}={(R\oplus R)}_{R}$ . Then, it is clear that$\text{Soc}M=\text{Soc}R\oplus \text{Soc}R$ which is essential inM_{R} and hence$M/\text{Soc}M$ has finite uniform dimension. Now, let$N=\text{Soc}R\oplus \text{Soc}R$ . Define${f}_{1}:M\to M$ by${f}_{1}(x,y)=(y,0)$ and${f}_{2}:M\to M$ by${f}_{2}(x,y)=(0,x)$ . Obviouslyf_{1} ,${f}_{2}\in \text{End}({M}_{R})$ . Let${N}_{1}=\text{Soc}R\oplus 0$ ,${N}_{2}=0\oplus \text{Soc}R$ . So, we have${f}_{1}({N}_{2})={N}_{1}\overline{)\subseteq}{N}_{2}$ and${f}_{2}({N}_{1})={N}_{2}\overline{)\subseteq}{N}_{1}$ . It follows thatM_{R} does not have pseudo duo property.
The following is a key lemma for our main theorem in this section.
Lemma 2.6.
Let
Now we have the following result which was pointed out in the introduction.
Theorem 2.7.
Let
Now assume that
Next we apply the former result to
Corollary 2.8.
Let
Recall that a module
Corollary 2.9.
Let
Corollary 2.10.
Let
We close this section by giving an example which illustrates that the converse of Theorem 2.7 is not true, in general.
Example 2.11.
Let
Now, let
3. Endomorphism Rings of Weak FI extending Modules
In this section our concern is the endomorphism ring of weak
Theorem 3.1.
Let
Next assume that
In any case, we have that
Corollary 3.2.
Let
Corollary 3.3.
Let
Since
Note that there are commutative, local rings
In the sense of construction certain examples, Corollary 3.3 is a useful tool. For example, let
Furthermore, the next example shows that the pseudo duo assumption in Theorem 3.1 is not unnecessary either.
Example 3.4.
Let
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