On Almost Projective Modules

Abstract

In this note, we investigate the relationship between almost projective modules and generalized projective modules. These concepts are useful for the study on the finite direct sum of lifting modules. It is proved that; if M is generalized N-projective for any modules M and N, then M is almost N-projective. We also show that if M is almost N-projective and N is lifting, then M is im-small N-projective. We also discuss the question of when the finite direct sum of lifting modules is again lifting.

1. Preliminaries and Introduction

Relative projectivity, injectivity, and other related concepts have been studied extensively in recent years by many authors, especially by Harada and his collaborators. These concepts are important and related to some special rings such as Harada rings, Nakayama rings, quasi-Frobenius rings, and serial rings.

Throughout this paper, R is a ring with identity and all modules considered are unitary right R-modules.

Lifting modules were first introduced and studied by Takeuchi [1]. Let M be a module. M is called a lifting module if, for every submodule N of M, there exists a direct summand K of M such that $N/K\ll M/K$. The lifting modules play an important role in the theory of (semi)perfect rings and modules with projective covers. The lifting module is not a generalization of projective modules. In fact, projective modules need not be lifting modules ${\mathbb{Z}}_{\mathbb{Z}}$. In general, direct sums of lifting modules are not lifting. $\mathbb{Z}/8\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}$ are lifting $\mathbb{Z}$-modules but $\mathbb{Z}/8\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$ is not lifting. This fact provides the motivation of this article.

Harada and Tozaki defined the concept of an almost projective module. Then they defined almost injective modules as a dual of almost projective modules. They gave a characterization of Nakayama rings in [2] by using almost projectivity. Let $M_1$ and $M_2$ be two modules. $M_1$ is called almost $M_2$-projective, if for every epimorphism $f:M_2\to X$ and every homomorphism $g:M_1\to X$, either there exists $h:M_1\to M_2$ with $fh=g$ or there exists a nonzero direct summand N of $M_2$ and a homomorphism $\gamma :N\to M_1$ with $g\gamma =f{\mid }_N$. If $M_1$ is almost $M_2$-projective for all finitely generated R-modules $M_2$, then $M_1$ is called almost projective. Baba and Harada proved that a module $M={\oplus }_{i=1}^n{M}_i$, where each $M_i$ is a hollow LE(local endomorphism) module is lifting if and only if $M_i$ is almost $M_j$-projective for $i\ne j$ and $i,j\in \{1.2,\cdots ,n\}$ in [3].

Let $\{M_i\mid i\in I\}$ be a family of modules. The direct sum decomposition $M={\oplus }_{i\in I}{M}_i$ is called to be exchangeable if, for any direct summand X of M, there exists $M_i^{\prime }\le M_i$ for every $i\in I$ such that $M=X\oplus (\oplus M_i^{\prime })$. A module M is called have (finite) internal exchange property if, any (finite) direct sum decomposition $M={\oplus }_{i\in I}{M}_i$ is exchangeable.

In [4], Mohamed and Müller defined generalized projectivity (dual of the concept of generalized injectivity) as follows. Let A and B be two modules. A is called generalized B-projective if, for any homomorphism $f:A\to X$ and any epimorphism $g:B\to X$, there exist decompositions $A=A_1\oplus A_2$ and $B=B_1\oplus B_2$, a homomorphism $h_1:A_1\to B_1$ and an epimorphism $h_2:B_2\to A_2$ such that $g{h}_1=f{\mid }_{A_1}$ and $f{h}_2=g{\mid }_{B_2}$. The generalized projectivity has roots in the study of direct sums of lifting modules. Kuratomi gave equivalent conditions for a module with exchange decomposition $M={\oplus }_{i=1}^n{M}_i$ to be lifting in terms of the relatively generalized projectivity of the direct summand of M in [5]. As a corollary, Kuratomi proved that finite direct sums of lifting modules are again lifting, when the distinct pairs of decomposition are relatively projective.

In [6], Alahmadi and Jain showed that generalized injectivity implies almost injectivity.

In this paper, we showed that generalized projectivity implies almost projectivity.

Result 1: 

Let M and N be right R-modules. If M is generalized N-projective, then M is almost N-projective.

Let M be any module. Consider the following conditions:

$\left(D_2\right)$ If $A\le M$ such that $M/A$ is isomorphic to a summand of M, then A is a summand of M.

$\left(D_3\right)$ If $M_1$ and $M_2$ are direct summands of M with $M=M_1+M_2$, then $M_1\cap M_2$ is a direct summand of M.

Then the module M is called discrete if it is lifting and satisfies the condition $\left(D_2\right)$ and it is called quasi-discrete if it is lifting and satisfies the condition $\left(D_3\right)$. Since $\left(D_2\right)$ implies $\left(D_3\right)$, every discrete module is quasi-discrete. In this paper, we give the relation between almost projective modules and some kind of generalized projective modules. We apply these results to a question when the finite direct sum of lifting module is lifting.

Result 2: 

Let M be a quasi-discrete module and N be a lifting module. If M is almost N-projective, M and N satisfy the descending chain conditions on direct summand, then M is strongly generalized epi-N-projective module.

Result 3: 

Let $M=M_1\oplus M_2$ be a module with finite internal exchange property. Assume that for any submodule A of M, if $M=A+M_2$, then $M\ne A+M_1$.Then the following are equivalent:

(1)

M is lifting.

(2)

$M_1$ and $M_2$ are lifting and for every decomposition $M=M_i\oplus M_j$, $M_i$ is generalized $M_j$-projective for $i\ne j$.

(3)

$M_1$ and $M_2$ are lifting and for every decomposition $M=M_i\oplus M_j$, $M_i$ is almost $M_j$-projective for $i\ne j$.

(4)

$M_1$ and $M_2$ are lifting and for every decomposition $M=M_i\oplus M_j$, $M_i$ is generalized small $M_j$-projective for $i\ne j$.

Result 4: 

Let $M_1,\cdots ,M_n$ be quasi-discrete and put $M=M_1\oplus \cdots \oplus M_n$. Then the followings are equivalent.

(1)

M is lifting with the (finite) internal exchange property,

(2)

M is lifting and the decomposition $M=M_1\oplus \cdots \oplus M_n$ is exchangeable,

(3)

$M_i$ is generalized $M_j$-projective for any $i\ne j\in \{1,\cdots ,n\}$.

(4)

$M_i\oplus M_j$ is lifting with the finite internal exchange property for $i\ne j\in \{1,\cdots ,n\}$,

(5)

$M_i$ is strongly generalized epi-$M_j$-projective and im-small $M_j$-projective for any $i\ne j\in \{1,\cdots ,n\}$,

(6)

$M_i$ is generalized epi-$M_j$-projective and im-small $M_j$-projective for any $i\ne j\in \{1,\cdots ,n\}$,

(7)

$M_i$ is strongly generalized epi-$M_j$-projective and almost $M_j$-projective for any $i\ne j\in \{1,\cdots ,n\}$,

(8)

$M_i$ is strongly generalized epi-$M_j$-projective and generalized small $M_j$-projective for any $i\ne j\in \{1,\cdots ,n\}$,

(9)

$M_i$ is generalized epi-$M_j$-projective and almost $M_j$-projective for any $i\ne j\in \{1,\cdots ,n\}$.

2. Almost Projectivity

In this section, we give the relation between generalized projective modules and almost projective modules.

Theorem 1.

Let M and N be right R-modules. If M is generalized N-projective, then M is almost N-projective.

Proof. 

Let $f:M\to X$ be any homomorphism and $g:N\to X$ be any epimorphism for any module X. By assumption there exist decompositions $N=N_1\oplus N_2$, $M=M_1\oplus M_2$, a homomorphism $h_1:M_1\to N_1$, and an epimorphism $h_2:N_2\to M_2$ such that $f{h}_2=g{\mid }_{N_2}$ and $g{h}_1=f{\mid }_{M_1}$. If f can not be lifted to N, then $N\ne N_1$. This means that $N_2\ne 0$. Define $h:N_2\to M$ with $h\left(n_2\right)=i_2{h}_2\left(n_2\right)$, where $i_2:M_2\to M$ is an inclusion map for $n_2\in N_2$. Now we will show that $fh=g{\mid }_{N_2}$. Take $n_2\in N_2$. $fh\left(n_2\right)=f\left(i_2\left(h_2\left(n_2\right)\right)\right)=f\left(h_2\left(n_2\right)\right)=g\left(n_2\right)$. Hence M is almost N-projective. □

Proposition 1.

Let $H_1,H_2,\cdots H_n$ be hollow modules and $M=H_1\oplus \cdots \oplus H_n$ exchangeable and N be a quasi-discrete module. If M is almost N-projective then M is generalized N-projective.

Proof. 

By the definition of almost projectivity, $H_i$ is also almost N-projective for all $i=1,2,\cdots ,n$. Clearly $H_i$ are generalized N-projective for all $i=1,2,\cdots ,n$. By [7] (Proposition 3.2), M is generalized N-projective. □

Now we will give the definitions of generalized epi projective modules and strongly generalized epi projective modules. Generalized epi projective modules were first defined in [8] under the name pseudo cojective modules and the authors gave the characterization of this module. Strongly generalized epi projective modules were first defined in [9].

Definition 1.

$M_1$ is (strongly) generalized epi-$M_2$-projective if, for any epimorphism $\phi :M_1\to X$ and any epimorphism $\pi :M_2\to X$, there exist decompositions $M=M_1^{\prime }\oplus M_1^{″}$, $M_2=M_2^{\prime }\oplus M_2^{″}$, a homomorphism (an epimorphism) ${\phi }_1:M_1^{\prime }\to M_2^{\prime }$ and an epimorphism ${\phi }_2:M_2^{″}\to M_1^{″}$ such that $\pi {\phi }_1=\phi {\mid }_{M_1^{\prime }}$ and $\phi {\phi }_2=\pi {\mid }_{M_2^{″}}$.

Clearly, if $M_1$ is strongly generalized epi-$M_2$-projective, then $M_1$ is generalized epi-$M_2$-projective for modules $M_1$ and $M_2$. To give the relation between almost projectivity and strongly generalized epi-projectivity of modules, we need to give some definitions. Let M be a module and let N and K be submodules of M with $N\subseteq K$. N is called a co-essential submodule of K in M if $K/N\ll M/N$ and it is denoted by $N{\subseteq }_{ce}K$ in M. Let X be a submodule of M. A is called a co-closed submodule in M if A does not have a proper co-essential submodule in M.

Theorem 2.

Let M be a quasi-discrete module and N be a lifting module. If M is almost N-projective, M and N satisfy the descending chain conditions on direct summand, then M is strongly generalized epi-N-projective module.

Proof. 

Let $f:M\to X$ and $g:N\to X$ be epimorphisms. Since M and N are lifting, there exist decompositions $M=M_1^{\prime }\oplus M_1^{″}$ and $N=N_1^{\prime }\oplus N_1^{″}$ such that $Kerf/M_1^{\prime }\ll M/M_1^{\prime }$ and $Kerg/N_1^{\prime }\ll N/N_1^{\prime }$. So we see that

$$f\left(M\right)=f\left(M_1^{\prime }\right)+f\left(M_1^{″}\right)=f\left(M_1^{″}\right),g\left(N\right)=g\left(N_1^{\prime }\right)+g\left(N_2^{″}\right)=g\left(N_2^{″}\right)$$

and

$$Ker\left(f{\mid }_{M_1^{″}}\right)=Kerf\cap M_1^{″}\ll M_1^{″},Ker\left(g{\mid }_{N_1^{″}}\right)=Kerg\cap N_1^{″}\ll N_1^{″}.$$

Thus we may assume that, $Kerf\ll M$ and $Kerg\ll N$. Since M is almost N-projective, then either there exists a homomorphism $h:M\to N$ such that $gh=f$ or there exists a decomposition of $N=N_1\oplus N_2$ and homomorphism $h_2:N_2\to M$ such that $f{h}_2=g{\mid }_{N_2}$. Consider the second case.

Since $N_2$ is lifting and M is amply supplemented, there exists a decomposition $N_2=N_2^{\prime }\oplus N_2^{″}$ such that $h_2\left(N_2^{\prime }\right)$ is coclosed in M and $h_2\left(N_2^{″}\right)\ll M$ by [9] (Lemma 1.6). Since M is lifting, $h_2\left(N_2^{\prime }\right)$ is a direct summand of M. Say $h_2\left(N_2^{\prime }\right)=M_2$. We also have

$$g\left(N\right)=g\left(N_1\right)+g\left(N_2\right)=g(N_1\oplus N_2^{\prime })+g\left(N_2^{″}\right)=g(N_1\oplus N_2^{\prime })+f{h}_2\left(N_2^{″}\right)=X.$$

Since $h_2\left(N_2^{″}\right)\ll M$, $f\left(h_2\left(N_2^{″}\right)\right)\ll X$ and hence we have $g\left(N\right)=g(N_1\oplus N_2^{\prime })$. Since $Kerg\ll N$, we have $N=N_1\oplus N_2^{\prime }$. This implies that $N_2^{″}=0$.

Since M is lifting, there exists a decomposition $M=K\oplus K^{\prime }$ such that $f^{-1}\left(g\left(N_1\right)\right)/K\ll M/K$. Since $N_1$ is coclosed in N, then $g\left(N_1\right)$ is coclosed in X. Since $K{\le }_{ce}{f}^{-1}\left(g\left(N_1\right)\right)\le M$ then $f\left(K\right){\le }_{ce}g\left(N_1\right)\le X$. This implies that $f\left(K\right)=g\left(N_1\right)$. We also have

$$f(f^{-1}\left(g\left(N_1\right)\right)+M_2)=f{f}^{-1}\left(g\left(N_1\right)\right)+f\left(M_2\right)=g\left(N_1\right)+g\left(N_2^{\prime }\right)=g\left(N\right)=X=f\left(M\right).$$

Since $Kerf\ll M$, $f^{-1}\left(g\left(N_1\right)\right)+M_2=M$. Then clearly $K+M_2=M$. Now we will show that $K\cap M_2=0$. By [7] (Lemma 1.7), $g\left(N_1\right)\cap g\left(N_2\right)\ll X$.

$$K\cap M_2\le f^{-1}\left(g\left(N_1\right)\right)\cap M_2\le f^{-1}(g\left(N_1\right)\cap g\left(N_2^{\prime }\right))=f^{-1}\left(g\left(N_1\right)\right)\cap f^{-1}\left(g\left(N_2^{\prime }\right)\right)\ll M.$$

Hence $K\cap M_2\ll M$. Since M is quasi-discrete, $K\cap M_2=0$.

Now we are in a position there exist decompositions $M=K\oplus M_2$, $N=N_1\oplus N_2$ and an epimorphism $h_2:N_2\to M_2$ with $f{h}_2=g{\mid }_{N_2}$ and $f\left(K\right)=g\left(N_1\right)$. By [2] (Proposition 4), K is almost $N_1$-projective, either there exists a decomposition of $N_1=T_1^{\prime }\oplus T_1^{″}$ and homomorphism $h_2^{\prime }:T_1^{\prime }\to K$ such that $f{h}_2^{\prime }=g{\mid }_{T_1^{\prime }}$ or there exists a homomorphism $h_1:K\to N_1$ such that $g{\mid }_{N_1}{h}_1=f$. If the first case hold, by the same manner of the above proof, we get $h_2^{\prime }$ is an epimorphism. If the second case hold, $g{h}_1\left(K\right)=f\left(K\right)=g\left(N_1\right)$ implies that $N_1=h_1\left(K\right)+Ker\left(g{\mid }_{N_1}\right)$. Since $N_1$ is lifting, we may assume that $Ker\left(g{\mid }_{N_1}\right)\ll N_1$. Then $h_1$ is an epimorphism. Since M and N satisfy descending chain conditions on direct summand, this process will stop. Hence we get M is strongly generalized epi-N-projective. □

Hence we can give an immediate result of Theorems 1 and 2.

Corollary 1.

Let M be a quasi-discrete module and N be a lifting module. If M is generalized N-projective, M and N satisfy the descending chain conditions on direct summand, then M is strongly generalized epi-N-projective module.

3. Generalized Small Projective Modules

In this section, we give the relation between generalized small projective modules and generalized projective modules. Generalized small projective modules were first defined in [8] as follows and the authors gave a characterization of this module.

Definition 2.

$M_1$ is generalized small $M_2$-projective if, for any homomorphism $\phi :M_1\to X$ with $Im\phi \ll X$ and any epimorphism $\pi :M_2\to X$, there exist decompositions $M=M_1^{\prime }\oplus M_1^{″}$, $M_2=M_2^{\prime }\oplus M_2^{″}$, a homomorphism ${\phi }_1:M_1^{\prime }\to M_2^{\prime }$ and an epimorphism ${\phi }_2:M_2^{″}\to M_1^{″}$ such that $\pi {\phi }_1=\phi {\mid }_{M_1^{\prime }}$ and $\phi {\phi }_2=\pi {\mid }_{M_2^{″}}$.

Now we will give the characterization of the generalized small projective module as follows:

Theorem 3

([8] Proposition 3.3). Let $M_1$ and $M_2$ be R-modules and $M=M_1\oplus M_2$. Then the following are equivalent:

(1) 

$M_1$ is generalized small $M_2$-projective.

(2) 

For every submodule A of M with $(A+M_1)/A\ll M/A$, there exists a decomposition $M=A^{\prime }\oplus M_1^{″}\oplus M_2^{\prime }=A^{\prime }+M_2$ such that $M_1^{″}\le M_1$, $M_2^{\prime }\le M_2$.

In general, generalized small projectivity does not imply generalized projectivity.

Example 1

([10] Example 2.7). Let S and $S^{\prime }$ be simple modules with $S\ncong S^{\prime }$ and let M and $K_1$ be uniserial modules such that $M\supset S\supset 0$, $K_1\supset K_2\supset S\supset 0$, $M/S\cong S$, $K_1/K_2\cong S$ and $K_2/S\cong S^{\prime }$. Then $K_1$ and M are lifting and $K_1$ is im-small M-projective. Hence $K_1$ is generalized small M-projective. But $K_1$ is not generalized M-projective.

Proposition 2.

Let K and L be any right R-modules. If K is generalized small-L-projective, then K is generalized small-$L^{*}$-projective for any direct summand $L^{*}$ of L.

Proof. 

Define $N=K\oplus L^{*}$. Let A be a submodule of N such that $(A+K)/A\ll N/A$. This implies that $(A+K)/A\ll M/A$. Since K is generalized small L-projective, there exists a decomposition $M=A^{\prime }\oplus K^{\prime }\oplus L^{\prime }=A^{\prime }+L$ such that $A^{\prime }\subseteq A$, $K^{\prime }\subseteq K$ and $L^{\prime }\subseteq L$. $N\cap M=N=N\cap (A^{\prime }\oplus K^{\prime }\oplus L^{\prime })=N\cap (A^{\prime }+L)$. Then we get $N=A^{\prime }\oplus K^{\prime }\oplus (N\cap L^{\prime })=A^{\prime }+N\cap L$. Since $N=K\oplus L^{*}$, $N\cap L=(K\oplus L^{*})\cap L=L^{*}\oplus (K\cap L)=L^{*}$. Then $N\cap L^{\prime }\subseteq N\cap L=L^{*}$. Clearly $A^{\prime }+N\cap L=A^{\prime }+L^{*}$. Then K is generalized small-$L^{*}$-projective. □

Proposition 3.

Let M be a lifting module with finite internal exchange property. Then for every decomposition $M=M_1\oplus M_2$, $M_i$ is generalized small $M_j$-projective for $i\ne j$ and $i,j\in \{1,2\}$.

Proof. 

It is obtained from [4] (Proposition 3.5). □

Proposition 4.

Let M be a quasi-discrete module. Then for every decomposition $M=M_1\oplus M_2$, $M_i$ is generalized small $M_j$-projective for $i\ne j$ and $i,j\in \{1,2\}$.

Proof. 

It is obtained by [11] (Proposition 4.23). □

Definition 3.

Let M and N be right R-modules. M is called im-small N-projective if for any submodule A of N, any homomorphism $f:M\to N/A$ with $Imf\ll N/A$ can be lifted to a homomorphism $g:M\to N$.

Now we give the relation between generalized small modules and im-small modules which is in [12] (Lemma 2.10). For the sake of completeness, we will give the proof of this lemma.

Lemma 1.

Let $M_1$ be any module and $M_2$ be a lifting module. If $M_1$ is generalized small $M_2$-projective, then $M_1$ is im-small $M_2$-projective.

Proof. 

Let $\pi :M_2\to X$ be an epimorphism and $\varphi :M_1\to X$ be a homomorphism with $Im\varphi \ll X$. Since $M_2$ is lifting, there exists a decomposition $M_2=A\oplus B$ such that $Ker\pi /B\ll M_2/B$. Then we have $\pi \left(M_2\right)=\pi \left(A\right)+\pi \left(B\right)=\pi \left(A\right)$. And we also have $Ker\pi {\mid }_A=ker\pi \cap A\ll A$. Hence we may assume that $Ker\pi \ll M_2$ by [5] (Proposition 2.1). Since $\pi :M_2\to X$ is a small epimorphism, for any submodule C of $M_2$, $\pi \left(C\right)\ll X$ if and only if $C\ll M_2$. Hence we cannot have a map from a direct summand of $M_2$ to $M_1$ satisfying the condition for $M_1$ to be generalized $M_2$-projective. Hence $M_1$ is im-small $M_2$-projective. □

Theorem 4.

Let M and N be any right R-modules. If M is an almost N-projective module and N is lifting, then M is im-small N-projective.

Proof. 

Let $g:N\to X$ be an epimorphism and let $f:M\to X$ be a homomorphism with $Imf\ll X$. Since N is lifting, we may assume that $Kerg\ll N$ as in the proof of Theorem 2. Since M is almost N-projective, there exists a homomorphism $h:M\to N$ such that $gh=f$ or there exists a decomposition of $N=N_1\oplus N_2$ and homomorphism $h_2:N_2\to M$ such that $f{h}_2=g{\mid }_{N_2}$. Consider the second case. Since $Imf\ll X$, $g\left(N_2\right)=fh\left(N_2\right)\ll X$. Then $g\left(N\right)=g\left(N_1\right)+g\left(N_2\right)=X$ implies that $g\left(N\right)=g\left(N_1\right)$. Since $Kerg\ll N$, $N=N_1$. Hence $N_2=0$. Therefore we have the first case. This completes the proof. □

Now we can give an immediate result of the Theorem 4, Theorem 1 and Lemma 1 as a generalization of [9] (Proposition 2.7).

Corollary 2.

Let M and N be lifting modules with the finite internal exchange property. Then M is generalized N-projective if and only if M is strongly generalized epi-N-projective and generalized small N-projective if and only if M is strongly generalized epi-N-projective and almost N-projective.

Lemma 2

([8] Lemma 4.9). Let $M_1$ and $M_2$ be modules and $M=M_1\oplus M_2$. Assume that for any submodule A of M if $M=A+M_2$, then $M\ne A+M_1$. If $M_1$ is generalized small $M_2$-projective, then $M_1$ is generalized $M_2$-projective.

Now we can apply this result when a finite direct sum of lifting modules is lifting.

Theorem 5.

Let $M=M_1\oplus M_2$ be a module with finite internal exchange property. Assume that for any submodule A of M, if $M=A+M_2$, then $M\ne A+M_1$.Then the following are equivalent:

(1) 

M is lifting.

(2) 

$M_1$ and $M_2$ are lifting and for every decomposition $M=M_i\oplus M_j$, $M_i$ is generalized $M_j$-projective for $i\ne j$.

(3) 

$M_1$ and $M_2$ are lifting and for every decomposition $M=M_i\oplus M_j$, $M_i$ is almost $M_j$-projective for $i\ne j$.

(4) 

$M_1$ and $M_2$ are lifting and for every decomposition $M=M_i\oplus M_j$, $M_i$ is generalized small $M_j$-projective for $i\ne j$.

Proof. 

$\left(1\right)\iff \left(2\right)\iff \left(4\right)$ They are clear [8] (Lemma 4.9).

$\left(2\right)\Rightarrow \left(3\right)$ It is clear by Theorem 1.

$\left(3\right)\Rightarrow \left(4\right)$ It is clear by definition and Theorem 4. □

Now we can give a result which is a generalization of [9] (Theorem 2.9).

Theorem 6.

Let $M_1$ and $M_2$ be lifting modules with the finite internal exchange property and put $M=M_1\oplus M_2$. Then the following are equivalent:

(1) 

M is lifting with the finite exchange property.

(2) 

M is lifting and the decomposition $M=M_1\oplus M_2$ is exchangeable.

(3) 

$M_1$ is generalized $M_2$-projective and $M_2$ is im-small $M_1$-projective.

(4) 

$M_2$ is generalized $M_1$-projective and $M_1$ is im-small $M_2$-projective.

(5) 

$\left(M_i\right)$ is strongly generalized epi-$M_j$-projective and im-small $M_j$-projective for $i\ne j$.

(6) 

$\left(M_i\right)$ is strongly generalized epi-$M_j$-projective and almost $M_j$-projective for $i\ne j$.

(7) 

$\left(M_i\right)$ is strongly generalized epi-$M_j$-projective and generalized small $M_j$-projective for $i\ne j$.

Proof. 

$\left(1\right)\iff \left(2\right)$ By [5] (Theorem 3.7).

$\left(2\right)\iff \left(3\right)\iff \left(4\right)\iff \left(5\right)$ By [9] (Theorem2.9).

$(5\iff (6)\iff (7)$ By Corollary 2. □

Theorem 7.

Let $M_1,\cdots ,M_n$ be quasi-discrete and put $M=M_1\oplus \cdots \oplus M_n$. Then the following are equivalent:

(1) 

M is lifting with the (finite) internal exchange property,

(2) 

M is lifting and the decomposition $M=M_1\oplus \cdots \oplus M_n$ is exchangeable,

(3) 

$M_i$ is generalized $M_j$-projective for any $i\ne j\in \{1,\cdots ,n\}$.

(4) 

$M_i\oplus M_j$ is lifting with the finite internal exchange property for $i\ne j\in \{1,\cdots ,n\}$,

(5) 

$M_i$ is strongly generalized epi-$M_j$-projective and im-small $M_j$-projective for any $i\ne j\in \{1,\cdots ,n\}$,

(6) 

$M_i$ is generalized epi-$M_j$-projective and im-small $M_j$-projective for any $i\ne j\in \{1,\cdots ,n\}$,

(7) 

$M_i$ is strongly generalized epi-$M_j$-projective and almost $M_j$-projective for any $i\ne j\in \{1,\cdots ,n\}$,

(8) 

$M_i$ is strongly generalized epi-$M_j$-projective and generalized small $M_j$-projective for any $i\ne j\in \{1,\cdots ,n\}$,

(9) 

$M_i$ is generalized epi-$M_j$-projective and almost $M_j$-projective for any $i\ne j\in \{1,\cdots ,n\}$.

Proof. 

$\left(1\right)\iff \left(2\right)\iff \left(3\right)\iff \left(4\right)\iff \left(5\right)\iff \left(6\right)$ follows by [9] (Theorem 2.16).

$\left(3\right)\iff \left(7\right)\iff \left(8\right)$ It is clear by Corollary 2.

$\left(3\right)\Rightarrow \left(9\right)$ It is clear by definition and Theorem 1.

$\left(9\right)\Rightarrow \left(6\right)$ It is clear by Theorem 4. □