### Article

Kyungpook Mathematical Journal 2020; 60(4): 673-682

**Published online** December 31, 2020

Copyright © Kyungpook Mathematical Journal.

### Direct Sums of Strongly Lifting Modules

Shahabaddin Ebrahimi Atani, Mehdi Khoramdel^{*} and Saboura Dolati Pishhesari

Department of Mathematics, University of Guilan, P. O. Box 1914, Rasht, Iran

e-mail : ebrahimi@guilan.ac.ir, mehdikhoramdel@gmail.com and saboura_dolati@yahoo.com

**Received**: June 20, 2019; **Revised**: August 5, 2020; **Accepted**: August 18, 2020

### Abstract

For the recently defined notion of strongly lifting modules, it has been shown that a direct sum is not, in general, strongly lifting. In this paper we investigate the question: When are the direct sums of strongly lifting modules, also strongly lifting? We introduce the notion of a relatively strongly projective module and use it to show if *M=M _{1}⊕ M_{2}* is amply supplemented, then

*M*is strongly lifting if and only if

*M*and

_{1}*M*are relatively strongly projective and strongly lifting. Also, we consider when an arbitrary direct sum of hollow (resp. local) modules is strongly lifting.

_{2}**Keywords**: lifting modules, strongly lifting modules, coclosed submodules, hollow modules, relatively strongly projective modules

### 1. Introduction

Supplemented and lifting modules are worthy of study in module theory since they are the duals of complemented and extending modules. A number of results concerning lifting modules have appeared in the literature in recent years. Lifting modules were first introduced by Takeuchi [11] but under the name codirect modules. An

It is of natural interest to investigate whether or not an algebraic notion for modules is inherited by direct summands and direct sums. The purpose of this paper is to study the direct sum of strongly lifting modules. The direct sum of two strongly lifting modules need not be strongly lifting. We look at when direct sums of finitely many strongly lifting modules are strongly lifting. It is shown that every strongly lifting module is a direct sum of hollow modules and every strongly lifting module is π-projective and a direct sum of hollow modules. We introduce the notion of relatively strongly projective modules and use it to show if _{1} ⊕ M_{2}_{1}_{2}

Throughout, all rings (not necessarily commutative rings) have identity and all modules are unital right modules. For completeness, we now state some definitions and notations used in this paper. Let ^{⊕}_{l}(R)_{r}(R)^{2} = e_{l}(End(M))_{l}(R)_{r}(R)_{1}_{2}_{1} ⊕ M_{2}

and _{t} M/K

The following are used in the sequel.

### Proposition 1.1.

(i)([13, Proposition 1.5]) Let

M be an amply supplemented module. Then every submodule ofM has an s-closure.(ii)([4, 3.7(6)]) Let

M be anR -module andK≤ L≤ M . If$K\stackrel{cc}{\hookrightarrow}M$ , then$K\stackrel{cc}{\hookrightarrow}L$ and the converse is true if$L\stackrel{cc}{\hookrightarrow}M$ .(iii)([13, 31.4(2)]) Let

M be an amply supplemented module andB ≤ C submodules ofM such thatC/B is co-closed inM/B andB is co-closed inM . ThenC is co-closed inM .(iv)([2, Lemma 1.1]) If

M=⊕ and_{i∈ I}M_{i}N is a fully invariant submodule of M, thenN=⊕ ._{i∈ I}(N∩ M_{i})

### Theorem 1.2. ([8, 15])

The following are equivalent for an

(1)

M is strongly lifting; (2) For each

N ≤ M , there existse∈ S such that_{l}(S)eM⊆ N and(1-e)M∩ N≪ (1-e)M .(3)

M is lifting and each direct summand of M is fully invariant. (4)

M is lifting and S is Abelian. (5)

M is amply supplemented and each coclosed submodule is a fully invariant direct summand in M .

### Proposition 1.3. ([8, 15])

Let

### Theorem 1.4. ([8, 15])

If

### Lemma 1.5. ([8, 15])

Let _{1} ⊕ M_{2}_{1}_{2}

### 2. Direct Sums of Strongly Lifting Modules

This section is devoted to investigate when direct sums of strongly lifting modules are strongly lifting. While it was shown in [8, 15] that every direct summand of a strongly lifting module is always strongly lifting, the following examples show that in general, the direct sum of strongly lifting modules is not a strongly lifting module.

### Example 2.1.

(1) Let

$R=\left(\begin{array}{cc}{\mathbb{Z}}_{2}& {\mathbb{Z}}_{2}\\ 0& {\mathbb{Z}}_{2}\end{array}\right)$ ,${M}_{1}=\left(\begin{array}{cc}{\mathbb{Z}}_{2}& {\mathbb{Z}}_{2}\\ 0& 0\end{array}\right)$ and${M}_{2}=\left(\begin{array}{cc}0& 0\\ 0& {\mathbb{Z}}_{2}\end{array}\right).$ ThenM and_{1}M are strongly lifting_{2}R -modules, howeverR=M is not strongly lifting, by Theorem 1.2._{1}⊕ M_{2}(2) Let

R be a uniserial ring andI a nonzero proper right ideal ofR . ThenR andI are strongly liftingR -modules. IfR ⊕ I is strongly lifting, thenR ⊕ I has SSSP. So it is a C3-module. This implies thatI≤ , by [7, Corollary 3.2], which is a contradiction.^{⊕}R

### Proposition 2.2.

(i) Let

M=⊕ . If_{i∈ I}M_{i}M is a strongly lifting module, thenHom(M for each_{i},M_{j})=0i≠ j ofI .

(ii) A free

R -moduleF is strongly lifting if and only ifR is strongly lifting and rank(_{R}F )=1.

(ii) Assume that

### Theorem 2.3.

(i) Let

M be anR -module. The following are equivalent:

(1)

M is strongly lifting;(2) If

M=N+K (N,K ≤ M ), then there exists a fully invariant direct summandX sayM=X ⊕ Y such thatX ⊆ N ,Y ⊆ K andY ∩ N ≪ Y .

(ii) Let

M be a strongly lifting module. ThenM is π-projective.

(ii) is clear from (i).

### Theorem 2.4.

Let _{i}

_{i}_{i}_{i}

Now, let _{i}

### Corollary 2.5.

Let

_{i}_{i}_{i}_{i}

By Corollary 2.5, we can see that every strongly lifting ring (considered as a module over itself) is semiperfect, the following example shows that the converse is not true, in general.

### Example 2.6.

Let

We now define a relative version of particular projective condition which is useful in our main theorems.

### Definition 2.7.

Let

This is obviously true if and only if _{i}_{j}

### Lemma 2.8.

(i) Let

M andN be two modules. IfM isN -strongly projective and${N}^{\prime}\le N$ , thenM is${N}^{\prime}$ -strongly projective and$N/{N}^{\prime}$ -strongly projective.(ii)

${\oplus}_{i\in I}{M}_{i}$ isN -strongly projective if and only ifM is_{i}N -strongly projective for eachi ∈ I .

(iii)

M is${\oplus}_{i=1}^{n}{N}_{i}$ -strongly projective if and only ifM isN -strongly projective for each_{i}1 ≤ i ≤ n .

(iv) Let

M be a finitely generated module. ThenM is⊕ -strongly projective if and only if_{i ∈ I}N_{i}M isN -strongly projective for each_{i}i∈ I .

(ii) Let _{i}

(iii) The necessity is clear by (i). For the sufficiency, it is sufficient to show that if _{1}_{2}_{1}⊕ N_{2}_{2}_{2}_{1}_{1}⊕ N_{2}

(iv) Assume that _{i}_{i}

The result of Lemma 2.8(iii) does not extend to infinite direct sums, as the following example shows.

### Example 2.9.

It is clear that

### Lemma 2.10.

Let

(1)

M is_{2}M -strongly projective;_{1}

(2) If

M=K+M for some_{1}K ≤ M , thenM ._{2}⊆ K

_{1}_{1}

In the following theorem, we present necessary and sufficient conditions under which direct sum of finite strongly lifting modules is strongly lifting.

### Theorem 2.11.

Let _{1} ⊕ M_{2}

(i)

M and_{1}M are relatively strongly projective._{2}

(ii)

M and_{1}M are strongly lifting._{2}

_{2}_{1}_{1}_{2}_{2}_{1}_{1}_{2}

Conversely, let $K$ be a coclosed submodule of $M$ such that $K+M_1=M$. By Lemma 2.10, _{1}_{1}_{1}_{2}=M

### Corollary 2.12.

Let _{i}_{i}

_{i}_{i}_{j}

Conversely, assume that _{i}

In [5], it is shown an _{2}(M)

### Theorem 2.13.

Let

### Corollary 2.14.

Let

In the next theorem, it is considered when direct sum of arbitrary hollow modules is strongly lifting.

### Theorem 2.15.

Let _{i}

(1)

M is strongly lifting;(2)

M is_{i}⊕ -strongly projective._{j ∈ I, j ≠ i}M_{j}

_{i}_{α}_{i}_{j}

In the following, it is considered when direct sum of an arbitrary local modules is strongly lifting.

### Corollary 2.16.

Let _{i}_{i}_{j}

_{i}_{i}_{j}

Conversely, let _{i}_{j}_{i}

In the next theorem, we use Theorem 2.15 to show that each factor module by a fully invariant submodule of a strongly lifting module is strongly lifting.

### Theorem 2.17

Let

_{i}_{i}_{j}

### Acknowledgements.

We would like to thank the referees for valuable comments.

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