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eISSN 0454-8124
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### Article

Kyungpook Mathematical Journal 2022; 62(4): 641-655

Published online December 31, 2022

### On Injectivity of Modules via Semisimplicity

Nguyen Thi Thu Ha

Faculty of Fundamental Science, Industrial University of Ho Chi Minh city, 12 Nguyen Van Bao, Go Vap District, Ho Chi Minh city, Vietnam
e-mail : nguyenthithuha@iuh.edu.vn

Received: March 11, 2022; Revised: April 19, 2022; Accepted: May 3, 2022

### Abstract

A right R-module N is called pseudo semisimple-M-injective if for any monomorphism from every semisimple submodule of M to N, can be extended to a homomorphism from M to N. In this paper, we study some properties of pseudo semisimple-injective modules. Moreover, some results of pseudo semisimple-injective modules over formal triangular matrix rings are obtained.

Keywords: semisimple-M-injective, soc-injective, Noetherian ring, V-ring

### 1. Introduction

Throughout the paper, R represents an associative ring with identity 1≠ 0 and all modules are unitary R-modules. We write MR (resp., RM) to indicate that M is a right (resp., left) R-module. We also write J (resp., Zr, Sr) for the Jacobson radical (resp., the right singular ideal, the right socle) of R and E(MR) for the injective hull of MR. If X is a subset of R, the right (resp., left) annihilator of X in R is denoted by rR(X) (resp., lR(X)) or simply r(X) (resp., l(X)) if no confusion appears. If N is a submodule of M (resp., proper submodule) we denote by N ≤ M (resp., N<M). Moreover, we write NeM,NM, NM and NmaxM to indicate that N is an essential submodule, a small submodule, a direct summand and a maximal submodule of M, respectively. A module M is called uniform if M≠ 0 and every non-zero submodule of M is essential in M.

Recently, some authors considered some generalizations of quasi-injective modules and automorphism-invariant modules (pseudo-injective modules)(see [1, 6, 9, 10, 12, 14, 15, 16, 17]). Some properties of automorphism-invariant modules and the structure of rings via the class of automorphism-invariant modules are studied (see [3, 8, 11, 18, 19]). In 2005, Hai Quang Dinh studied a generalization of the M-injective module that is pseudo M-injective. A module N is called pseudo M-injective if for any submodule A of M and every monomorphism from A to N, can be extended to a homomorphism from M to N. A module M is called pseudo-injective if M is pseudo M-injective.

A generalization of M-injective modules, Amin-Yousif-Zeyada ([4]) introduced the soc M-injective. A right R-module N is called soc-M-injective if for any homomorphism Soc(M)N, can be extended to a homomorphism from M to N. A module M is called soc-quasi-injective if M is soc-M-injective.

The purpose of this paper, we consider a generalization of soc-M-injective and pseudo M-injective modules, that is pseudo semisimple-M-injective. We call that a module N is pseudo semisimple-M-injective if for any monomorphism from every semisimple submodule of M to N, can be extended to a homomorphism from M to N. A module M is called pseudo semisimple-injective if M is pseudo semisimple-M-injective. In this paper, we will give some properties of pseudo semisimple-injective modules and structure of rings via these modules.

In Section 2, we give some basic properties of pseudo semisimple-injective modules and relatively pseudo semisimple-injective modules. It is well known that a module pseudo-injective is direct-injective (C2-module) (see [6, Theorem 2.6]). We study this result for pseudo semisimple-injective modules. We prove in Proposition 2.4 that pseudo semisimple-injective modules are semisimple-direct-injective. On the other hand, we show that if M=iIMi is a direct sum of uniform submodules Mi, then M is soc-quasi-injective if and only if M is pseudo semisimple-injective (see Theorem 2.12). Next, we consider the projectivity of socles of modules via the pseudo semisimple-injectivity and we obtain in Theorem 2.13 that; if M is a projective module, then Soc(M) is projective iff every quotient module of a pseudo semisimple-M-injective module is pseudo semisimple-M-injective, iff every quotient module of a semisimple-M-injective module is pseudo semisimple-M-injective, iff every quotient module of an injective module is pseudo semisimple-M-injective. From the definition of pseudo semisimple-injective module, we study structure of rings in which every semisimple right module is pseudo semisimple-M-injective for every cyclic rightmodule M. We show that a ring R is a right Noetherian right V-ring iff every semisimple right R-module is pseudo semisimple-M-injective for every cyclic right R-module M, iff every right R-module is pseudo semisimple-M-injective for every cyclic right R-module M (see Theorem 2.23). Some other properties are studied and extended. Finally, we study the pseudo semisimple-injectivity of modules over formal triangular matrix rings.

### 2. On pseudo semisimple-injective modules

Definition 2.1. A right R-module N is called pseudo semisimple-M-injective if for any semisimple submodule A of M, any monomorphism f:AN extends to a homomorphism from M to N. A module M is called pseudo semisimple-injective if M is pseudo semisimple-M-injective.

A right R-module N is called soc-M-injective if for any homomorphism from Soc(M) to N, can be extended to a homomorphism from M to N. A module M is called soc-quasi-injective if M is soc-M-injective (see [4]).

All soc-M-injective modules are pseudo semisimple-M-injective. But the converse is not true in general.

Example 2.2. Assume that a right R-module M has only five submodules 0,M1,M2, M1M2,M, which M1M2 and End(Mi)2 (see Hallett's example and Teply's example). Then M is pseudo semisimple-M-injective. Note that Soc(M)=M1M2 and the projection of Soc(M) to M1 cannot be extended to a homomorphism from M to M. It follows that M is not soc-M-injective.

Lemma 2.3. Let M and N be two modules.

• (1) If N is pseudo semisimple-M-injective and A is a direct summand of N, then A is pseudo semisimple-M-injective.

• (2) If N is pseudo semisimple-M-injective and B is a closed submodule of M, then N is pseudo semisimple-B-injective.

• (3) If M is pseudo semisimple-injective, then A is pseudo semisimple-injective for all fully invariant closed submodule A of M.

Proof. It is obvious.

A module M is called semisimple-direct-injective if for any semisimple submodules A, B of M with A≅ B and B a direct summand of M, A is a summand of M (see [2]).

Proposition 2.4. Every pseudo semisimple-injective module is semisimple-direct-injective.

Proof. Assume that M is a pseudo semisimple-injective module. Let B be a direct summand of M and A be a semisimple submodule of M with AB. We show that B is a direct summand of M. Let f:AB be an isomorphism. We have that B is a direct summand of M and obtain that B is pseudo semisimple-M-injective by Lemma 2.3. There exists a homomorphism α:MB that is an extension of f.

That is αι=f with the inclusion map ι:AM. We deduce that ι splits and so A is a direct summand of M.

Corollary 2.5. Let M be a pseudo semisimple-injective module. If M=A1A2 where A1 is semisimple and f:A1A2 is a homomorphism, then Im(f) is a direct summand of A2.

Theorem 2.6. Let R and S be Morita-equivalent rings with the category equivalence F:ModRModS. Let M, N and K be right R-modules and f:HL be a homomorphism of right R-modules. Then:

• (1) KR is semisimple if and only if F(K)S is semisimple.

• (2) f is a monomorphism if and only if F(f) is a monomorphism.

• (3) MR is pseudo semisimple-N-injective if and only if F(M)S is pseudo semisimple-F(N)S-injective.

Proof (1) and (2) by [5, Proposition 21.4, 21.8].

(3) is followed from (1) and (2).

A ring R is called right pseudo semisimple-injective if RR is pseudo semisimple-injective.

Corollary 2.7. Right pseudo semisimple-injectivity is a Morita invariant property of rings.

Proposition 2.8. Let M and N be modules and X=MN. The following conditions are equivalent:

• (1) N is soc-M-injective.

• (2) For each semisimple submodule K of X, where KN=0, there exists C≤ X such that KC and NC=X.

Proof. (1)(2). Let K be a semisimple submodule of X, with KN=0, πM:MNM and πN:MNN the canonical projections. We can check that NK=NπM(K) and πM(K) is a semisimple submodule of M. Let φ:πM(K)πN(K) be a homomorphism defined as follows: for k=m+nK (with mM,nN), φ(m)=n. It is easy to see that φ is a monomorphism. Since N is pseudo semisimple-M-injective, there is a homomorphism φ¯:MN, which extends φ. Let C={mφ¯(m)|mM} be a submodule of X. Then X=NC and K is contained in C.

(2)(1). Let A be a semisimple submodule of M and φ:AN be a homomorphism. Put K={aφ(a)|aA} be a submodule of X. It follows that KAφ(A). Then πM(K)=A, NK=NπM(K)=NA and K is a semisimple submodule of X. By assumption, there exists a submodule C of X containing K with NC=X. Let π:NCN be the natural projection. Then the restriction π|M extends φ, proving (1).

Similarly, we have a result for pseudo semisimple-M-injective modules.

Proposition 2.9.Let M and N be modules and X=MN. The following conditions are equivalent:

• (1) N is pseudo semisimple-M-injective.

• (2) For each semisimple submodule K of X, where KM=KN=0, there exists C≤ X such that K≤ C and NC=X.

Theorem 2.10. If MN is a pseudo semisimple-injective module, then N is soc-M-injective.

Proof. Assume that MN is pseudo semisimple-injective, and f:Soc(M)N is a homomorphism. Define g:Soc(M)MN by g(m)=(m,f(m)) (for all mSoc(M)). Clearly, g is a monomorphism. By Lemma 2.3, MN is pseudo semisimple-M-injective, whence g extends to a homomorphism g*:MMN. Let π:MNN be the natural projection. Then πg* is a homomorphism extending f. Consequently, N is soc-M-injective.

Corollary 2.11. For any integer n2, Mn is pseudo semisimple-injective if and only if M is soc-quasi-injective.

Theorem 2.12. Let M=iIMi be a direct sum of uniform submodules Mi. Then M is soc-quasi-injective if and only if M is pseudo semisimple-injective.

Proof. () is obvious.

() First let M be a uniform pseudo semisimple-injective module. Let f:Soc(M)M be a homomorphism. If Kerf=0, then f can be extended to an endomorphism of M. Otherwise, Kerf0. Let g=ιf, where ι:Soc(M)M is the inclusion homomorphism. Since Kerf0 and M is uniform, Kerg=0. Then, by the pseudo semisimple-injectivity, g can be extended to some hEnd(M). Now 1MhEnd(M) is an extension of f. Thus M is soc-quasi-injective.

Now let M be a pseudo semisimple-injective module and M=iIMi. For all j∈ I, we have iI\{j}Mi is pseudo semisimple-Mj-injective by Theorem 2.10. Since direct summands of pseudo semisimple-injective are obviously pseudo semisimple-injective and by the remark above, each Mj is soc-quasi-injective. Therefore, M is soc-quasi-injective

Theorem 2.13. The following conditions are equivalent for a projective module M:

• (1) Soc(M) is projective.

• (2) Every quotient module of a pseudo semisimple-M-injective module is pseudo semisimple-M-injective.

• (3) Every quotient module of a soc-M-injective module is pseudo semisimple-M-injective.

• (4) Every quotient module of an injective module is pseudo semisimple-M-injective.

Proof. (1)(2). Assume that ER is pseudo semisimple-M-injective and π:EB is an epimorphism. Let f:SB be a monomorphism with S a semismple submodule of M.

where ι is the inclusion.

By (1), Soc(M) is projective, and so S is projective. Therefore, there exists an R-homomorphism h:SE such that πh=f. Since f is monomorphism, h is too. Now since E is pseudo semisimple-M-injective, there is an R-homomorphism h:ME such that hι=h. Let h=πh:MB, then hι=f. This means B is pseudo semisimple-M-injective.

(2)(3)(4) is obvious.

(4)(1). We consider the epimorphism h:AB and an R-homomorphism α:Soc(M)B.

Since B=h(A)ψA/Kerhι1E(A)/Kerh, where ι1 is the inclusion and ψ(h(a))=a+Kerh, for all aA. Then let j=ι1ψ. We consider the following diagram:

Soc(M)ιMφαAhB0jE(A)pE(A)/Kerh0

where ι is the inclusion and p is the natural epimorphism.

By (4), E(A)/Kerh is pseudo semisimple-M-injective and then there exists an R-homomorphism α:ME(A)/Kerh such that αι=jα. Since M is projective, there is an R-homomorphism α:ME(A) such that pα=α. Let h=αι:Soc(M)E(A). It is easy to see that h(Soc(M))A, so there exists an R-homomorphism φ:Soc(M)A such that φ(x)=h(x), for all xSoc(M).

Now we claim that hφ=α. In fact, for each xSoc(M) we have

j(α(x))=α(ι(x))=α(x)=p(α(x))=p(h(x))=p(φ(x)).

Since α is an epimorphism, α(x)=h(a) for some aA. Therefore j(α(x))=j(h(a))=a+Kerh, and so a+Kerh=φ(x)+Kerh, h(aφ(x))=0. Hence hφ(x)=h(a)=α(x). Thus Soc(M) is projective.

Corollary 2.14. The following conditions are equivalent:

• (1) Soc(RR) is projective.

• (2) Every quotient module of a pseudo semisimple-RR-injective module is pseudo semisimple-RR-injective.

• (3) Every quotient module of a soc-RR-injective module is pseudo semisimple-RR-injective.

• (4) Every quotient module of an injective module is pseudo semisimple-RR-injective.

Proposition 2.15. Let M be a finitely generated module. If every direct sum of pseudo semisimple-M-injective modules is pseudo semisimple-M-injective, then Soc(M) is finitely generated.

Proof. Assume that Soc(M)=ISi with Si simple. Let i:Soc(M)IE(Si) be the inclusion monomorphism. Since IE(Si) is pseudo semisimple-M-injective, there exists a homomorphism g:MIE(Si) such that g is an extension of i. Since M is finitely generated, i(Soc(M))=g(Soc(M))KE(Si) for some finite subset K of I. Moreover, Soc(KE(Si)) is finitely generated and so Soc(M) is finitely generated.

Proposition 2.16. For a right R-module M, the following conditions are equivalent:

• (1) M is soc-E(M)-injective.

• (2) M is pseudo semisimple-N-injective for all right R-modules N.

Proof. (1)(2) by [4, Theorem 3.1].

(2)(1). By [4, Theorem 3.1], we only prove M=ET with E injective and Soc(T)=0. If Soc(M)=0, we are done. Otherwise, we have that M is pseudo semisimple-E(Soc(M))-injective and obtain that there exists a homomorphism f:E(Soc(M))M such that f(x)=x for all x∈ Soc(M). Since Soc(M)eE(Soc(M)), f is a monomorphism. That means f is a splitting monomorphism. Thus, M=ET with E injective and Soc(T)=0.

Corollary 2.17. The following conditions on a ring R are equivalent:

• (1) R is right Noetherian.

• (2) If S1,S2,,Sn are simple right R-modules, i=1E(Si) is pseudo semisimple-N-injective for all right R-modules N.

Lemma 2.18. The following conditions are equivalent for a right R-module M:

• (1) Every right R-module is pseudo semisimple-M-injective.

• (2) Every semisimple right R-module is pseudo semisimple-M-injective.

• (3) Soc(M) is a direct summand of M.

Proof. (1)(2) and (3)(1) are obvious.

(2)(3). Assume that every semisimple right R-module is pseudo semisimple-M-injective. Then, Soc(M) is pseudo semisimple-M-injective. It follows that Soc(M) is a direct summand of M.

A ring R is called a right V-ring if every simple right R-module is injective.

Proposition 2.19. The following conditions are equivalent for a ring R:

• (1) R is a right V-ring.

• (2) Every finitely cogenerated right R-module is a pseudo semisimple-injective right R-module.

Proof (1)(2) is obvious.

(2)(1). Let S be a simple right R-module. Then, SE(S) is a finitely cogenerated R-module. Take ι:SE(S) the inclusion map. It follows that S=ι(S) is a direct summand of E(S) by Corollary 2.5. We deduce that E=E(S) is injective.

Corollary 2.20. The following conditions are equivalent for a ring R:

• (1) R is a right Noetherian right V-ring.

• (2) SE(S) is a pseudo semisimple-injective right R-module for all simple right R-module S.

Similarly, we also have the following result for Noetherian V-rings.

Proposition 2.21. The following conditions are equivalent for a ring R:

• (1) R is a right Noetherian right V-ring.

• (2) Every right R-module with essential socle is a pseudo semisimple-injective right R-module.

Proof. (1)(2) is obvious.

(2)(1). Let {Si}iI be a family of simple modules. Then, (iISi)E(iISi) is a right R-module with essential socle, and so it is a semisimple-injective right R-module. It follows that iISi is injective.

Corollary 2.22. The following conditions are equivalent for a ring R:

• (1) R is a right Noetherian right V-ring.

• (2) SE(S) is a pseudo semisimple-injective right R-module for all semisimple right R-module S.

Theorem 2.23. The following conditions are equivalent for a ring R:

• (1) R is a right Noetherian right V-ring.

• (2) Every semisimple right R-module is pseudo semisimple-M-injective for every cyclic right R-module M.

• (3) Every right R-module is pseudo semisimple-M-injective for every cyclic right R-module M.

Proof. (1)(2). Since R is a right Noetherian right V-ring, every semisimple right R-module is injective, and hence every semisimple right R-module is pseudo semisimple-M-injective for every cyclic right R-module M.

(2)(3). Assume that every semisimple right R-module is pseudo semisimple-C-injective for every cyclic right R-module C. Let M be a cyclic right R-module. Then, Soc(M) is a direct summand of M. We deduce that every right R-module is pseudo semisimple-M-injective by Lemma 2.18.

(3)(1) We show that every semisimple right R-module is injective. Let S be a semisimple right R-module and N be a cyclic right R-module. Then, every right R-module is pseudo semisimple-N-injective by (3). It follows that Soc(N) is a direct summand of N by Lemma 2.18. This implies that S is semisimple-N-injective. We deduce that S is injective by [4, Lemma 3.11].

Enochs [7] introduced the injective cover notion which is the dual to the injective envelope, and showed that a ring R is a right Noetherian ring if and only if every right R-module has an injective cover. Now, we introduce the pseudo semisimple-injective cover notion.

Definition 2.24. An R-homomorphism g:EM is called a psi-cover of a right R-module M if E is a pseudo semisimple-injective module such that any diagram

with E' a pseudo semisimple-injective module can be completed; and the diagram

can be completed only by an automorphism α.

Now, we prove in Theorem 2.25 that a ring R is a right Noetherian right V-ring if and only if every right R-modules with essential socle has a psi-cover.

Theorem 2.25. The following are equivalent for a ring R:

• (1) R is a right Noetherian right V-ring.

• (2) Every right R-modules with essential socle has a psi-cover.

Proof. (1)(2). It is obvious.

(2)(1) Let S be a semisimple right R-module and let M=SE(S). We show that M is pseudo semisimple-injective. Call g:EM a psi-cover of M. Consider the following diagrams:

where ι1:MS and ι2:ME(S) are the canonical injections. Note that all modules S and E(S) are pseudo semisimple-injective modules. By the definition of psi-cover, there exist homomorphisms α1:SE and α2:E(S)E such that gαi=ιi for i=1,2. Define α:ME by α(x1+x2)=α1(x1)+α2(x2) for all x1S and x2E(S). It can easily be checked that α is well-defined and we have

gα(x1+x2)=gα1(x1)+gα2(x2)=ι1(x1)+ι2(x2)=x1+x2.

Thus, gα=1M, and α:ME is a split monomorphism. Then M is isomorphic to a direct summand of E. Since a direct summand of a pseudo semisimple-injective module is again a pseudo semisimple-injective module, M is a pseudo semisimple-injective module. By Corollary 2.22, R is a right Noetherian V-ring.

Let R and S be two rings and M be an R-S-bimodule (left R-module and right S-module). Take

K=RM0S={rm0s|rR,sS,mM}

a ring with the addition and multiplication as follows:

rm0s+ r m0 s=r+r m+m 0s+s
rm0s r m0 s=rr rm +ms0ss

The ring K is also called a formal triangular matrix ring (see [13]). It is well-known that the category of right K-module Mod-K is equivalent to the category T of triples (X, Y, f), where X is a right R-module, Y is a right S-module and f:XRMY is a homomorphism of right S-modules. The right K-module (X, Y, f) is the additive group XY with right K-action given by

(xy)rm0s=(xr,f(xm)+ys)

Next, we consider homomorphisms of K-modules. Let (X1,Y1,f1) and (X2,Y2,f2) be right K-modules. A right K-homomorphism φ:(X1,Y1,f1)(X2,Y2,f2) is a pair (φ1,φ2) where φ1:X1X2 is a homomorphism of right R-modules and φ2:Y1Y2 is a homomorphism of right S-modules such that the following diagram is commutative

Note that a K-homomorphism φ=(φ1,φ2):(X1,Y1,f1)(X2,Y2,f2) is a monomorphism (epimorphism) if and only if φ1 and φ2 are monomorphisms (epimorphisms).

A submodule of a right K-module (X,Y,f) is a triple (X0,Y0,f0), where X0XR, Y0YS such that the following diagram is commutative.

with ι1:X0X, ι2:Y0Y the inclusion maps.

Proposition 2.26. Let K=RM0S and (X,Y,f) be a right K-module. If (X,Y,f) is a pseudo semisimple-injective right K-module then

• (1) Y is a pseudo semisimple-injective right S-module.

• (2) H={xX|f(xm)=0for all mM} is a pseudo semisimple-injective right R-module.

Proof. (1) Let Y0 be a semisimple submodule of Y and φ:Y0Y is an S-monomorphism. Then, (0,Y0,0) is a semisimple submodule of K-module (X,Y,f) and γ=(0,φ):(0,Y0,0)(X,Y,f) is a K-homomorphism. By our assumption, (0,φ) is a K-monomorphism, and so there exists an endomorphism θ=(θ1,θ2) of (X,Y,f) such that 𝜃 is an extension of γ. It follows that θ2:YY is an extension of φ. Hence Y is a pseudo semisimple-injective module.

(2) Let X0 be a semisimple submodule of H and β:X0H is an R-monomorphism. Then, (X0,0,0) is a semisimple submodule of K-module (X,Y,f) and δ=(β,0):(X0,0,0)(X,Y,f) is a K-monomorphism, and so there exists an endomorphism ω=(ω1,ω2) of (X,Y,f) such that ω is an extension of δ. It means that the following is commutative

and so, ω2f=f(ω11M). We deduce that ω1(H)H. Then, ω1|H:HH is an extension of β. It shows that H is a pseudo semisimple-injective module.

Proposition 2.27. Let K=RM0S and (X,Y,f) be a right K-module. If

• (1) Y is a pseudo semisimple-injective right S-module and

• (2) H={xX|f(xm)=0for all mM} is a pseudo semisimple-injective right R-module.

then (H,Y,0) is a pseudo semisimple-injective right K-module.

Proof. Let (X0,Y0,f0) be a semisimple submodule of (H,Y,0) and α=(α1,α2):(X0,Y0,f0)(H,Y,0) is a K-monomorphism. Then, f0=0 and α1:X0H, α2:Y0Y are monomorphisms. Note that X0 is a semisimple submodule of H and Y0 is a semisimple submodule of Y. Since H and Y are pseudo semisimple-injective, there exist an endomorphism β1 of H and β2 of Y such that β1 is an extension of α1 and β2 is an extension of α2. One can check that β=(β1,β2) is an endomorphism of (H,Y,0) and it is an extension of α.

Let (X,Y,f) be a right K-module. Then, we have the following R-homomorphism

f˜:XHomS(M,Y)xf˜(x):MYmf˜(x)(m)=f(xm)

Proposition 2.28. Let K=RM0S and (X,Y,f) be a right K-module. If

• (1) Y is a pseudo semisimple-injective right S-module and

• (2) f˜ is an isomorphism of right R-module.

then (X,Y,f) is a pseudo semisimple-injective right K-module.

proof. Let (X0,Y0,f0) be a semisimple submodule of (X,Y,f) and α=(α1,α2):(X0,Y0,f0)(X,Y,f) is a K-monomorphism. Then, α1:X0X and α2:Y0Y are monomorphisms with α2f0=f(α11M). Note that Y0 is a semisimple submodule of Y. Since Y is a pseudo semisimple-injective module, there exists an endomorphism β2 of Y such that β2 is an extension of α2.

Fix xX. For any mM, set θ(m)=β2(f(xm)). It follows that θ:MY is an S-homomorphism. By assumption there exists a unique element xX such that f˜(x)=θ. Then, for all mM we have

f(xm)=f˜(x)(m)=θ(m)=β2(f(xm))

We define β1:XX via β1(x)=x. One can check that β1 is an R-homomorphism and satisfies f(β11M)=β2f. This means that β=(β1,β2):(X,Y,f)(X,Y,f) is a K-homomorphism. Next, we show that β1 extends α1. In fact, for any x0X0 and for all mM, we have (α2f0)(x0m)=f(α11M)(x0m) or β2f(x0m)=f(α1(x0)m). It follows that f(β1(x0)m)=f(α1(x0)m) or f˜(β1(x0))=f˜(α1(x0)). Since f˜ is an isomorphism, β1(x0)=α1(x0). We deduce that β extends α and so, (X,Y,f) is pseudo semisimple-injective.

### Footnote

The researcher wish to express our deep sense of gratitude to Industrial University of Ho Chi Minh City for the financial support offered to this research project according to the Scientific Research Contract No 59/HD-DHCN, code 21.2CB01.

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