### Article

Kyungpook Mathematical Journal 2020; 60(1): 44-51

**Published online** March 31, 2020

Copyright © Kyungpook Mathematical Journal.

### On Generalized FI-extending Modules

Canan Celep Yücel

Department of Mathematics, Faculty of Arts and Sciences, Pamukkale University, Denizli 20070, Turkey

e-mail : ccyucel@pau.edu.tr

**Received**: June 3, 2016; **Revised**: November 2, 2019; **Accepted**: December 3, 2019

### Abstract

A module

**Keywords**: fully invariant submodule, FI-extending, GFI-extending.

### 1. Introduction

In recent years, the theory of extending modules and rings has come to play an important role in the theory of rings and modules. A module _{11}

In [2], the following statements were proved: (1) Any direct sum of FI-extending modules is FI-extending; (2) A ring _{n}

In this work, we determine a generalization of the FI-extending module which is not only preserved under various extensions including direct sums and matrix constructions. We define a module _{n}

Throughout this paper, all rings are associative with unity and all modules are unital right modules. Recall from [2], a submodule _{R}_{R}_{n}

### 2. GFI-extending Modules

In this section, we begin with the definition of the main concept of this paper. Then, we study relationships between the extending, FI-extending and GFI-extending modules. We also consider connections between the singular and GFI-extending modules.

### Definition 2.1

Let

### Proposition 2.2

M is extending ,M is FI-extending ,M is GFI-extending.

**Proof**

(i) ⇒ (ii) and (ii) ⇒ (iii). These implications are clear.

Let _{R}_{R}_{R}_{R}

Finally, let _{2}[_{1}, _{2}] be the commutative polynomial ring with the indeterminants _{1}, _{2} over the field ℤ_{2} and _{1}_{2}. Then

### Lemma 2.3

**Proof**

This proof is routine.

In general, the reverse implication of the above result does not hold. For example, let ℤ be the ring of all integers. Then, ℤ is extending. Hence, it is FI-extending. By Proposition 2.2, ℤ is GFI-extending as a right ℤ-module, but it is nonsingular.

The following lemma gives the equivalence of the FI-extending and GFI-extending modules.

### Lemma 2.4

**Proof**

Let

The following proposition is a consequence of Definition 2.1 and gives a characterization of the GFI-extending modules by fully invariant submodules.

### Proposition 2.5

M is GFI-extending. For any fully invariant submodule N of M, M has a decomposition M =K ⊕K′ such that N ≤K and that M/ (K′ +N )is singular. For any fully invariant submodule N of M, M/N has a decomposition M/N =K/N ⊕K′/N such that K is a direct summand of M and that M/K′ is singular. For any fully invariant submodule N of M, there is a direct summand K of M such that for any x ∈K there is an essential right ideal I of R such that xI ≤N.

**Proof**

The proof is straightforward.

### Lemma 2.6

**Proof**

Assume

### Proposition 2.7

**Proof**

Let _{R}_{R}_{R}_{R}_{R}_{e} R_{R}

The next result establishes connections between a GFI-extending module and its injective hull.

### Proposition 2.8

_{R}^{2} = _{R}

**Proof**

(⇒). Assume that _{R}

(⇐). Let

In the next lemma and theorem, we prove that the GFI-extending property of a ring _{n}

### Lemma 2.9

^{2} = _{n}_{n}

**Proof**

The proof is a clear consequence of [2, Lemma 2.2] because _{n}_{n}_{R}_{n}_{n}_{n}_{n}

### Theorem 2.10

_{n}

**Proof**

Let _{n}_{n}^{2} ∈ _{n}_{n}_{n}

### 3. Direct Sum of GFI-Extending Modules

In [2], it was proved that a direct sum of FI-extending modules is FI-extending. It is also known that a direct sum of singular modules is singular [7]. In this section, we show that GFI-extending property is closed under direct sums.

### Theorem 3.1

**Proof**

Suppose the modules _{i}_{i}_{∈}_{I}A_{i}_{i}_{∈}_{I}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}/_{i}_{i}_{∈}_{I}H_{i}

### Corollary 3.2

### Corollary 3.3

M is finitely generated. M is of bounded order (i.e., nM = 0,for some positive integer n). M is divisible.

**Proof**

(i) Every finitely generated Abelian group is a direct sum of uniform ℤ-modules.

(ii) This part is from [10, p.262].

(iii) If

The following corollaries are direct consequences of Theorem 3.1.

### Corollary 3.4

_{1} ⊕ _{2}_{1}_{2}

**Proof**

The proof is clear by Theorem 3.1.

### Corollary 3.5

_{1} ⊕ _{2}_{1}_{2}_{1}

**Proof**

Let _{1} is a direct summand of _{1}) ⊕ _{1} is a fully invariant submodule of _{1}. Since _{1} is GFI-extending, there exists a direct summand _{1} such that _{1}) is singular. But, _{1}) ⊕ _{1} = _{1} ⊕ _{1}) ⊕ _{1}) ⊕ _{1} and hence a direct summand of

In general, it is not known whether a direct summand of a GFI-extending module is also GFI-extending. However, in the following theorem, we prove that this implication is true under some restrictions.

### Theorem 3.6

**Proof**

By Lemma 2.6,

### Proposition 3.7

**Proof**

If _{R}

### Acknowledgements

The author gratefully thanks to the Referee for the constructive comments and recommendations which definitely help to improve the readability and quality of the paper.

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