EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 3, 2018, 815-822
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

On Slightly Compressible-Injective Modules

Nguyen Dang Hoa Nghiem1, Phatsarapa Janmuang1, Samruam Baupradist1,3,∗,
Ronnason Chinram2

1 Department of Mathematics and Computer Science, Faculty of Science,
Chulalongkorn University, Bangkok, 10330, Thailand
2 Algebra and Applications Research Unit, Department of Mathematics and Statistics,
Faculty of Science, Prince of Songkla University, Hat Yai, Songkhla, 90110, Thailand
3 Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok 10400, Thailand

Abstract. In this paper, we introduce the concept of slightly compressible-injective modules,
following this, a right R-module N is called an M -slightly compressible-injective module, if every
R-homomorphism from a non-zero M -slightly compressible submodule of M to N can be extended
to M . We give some characterizations and properties of slightly compressible-injective modules.

2010 Mathematics Subject Classifications: 16D50, 16D70, 16D80

Key Words and Phrases: M -slightly compressible modules; M -slightly compressible-injective
modules; quasi-slightly compressible-injective modules.

1. Introduction

Throughout all rings are associative with identity and modules are unitary right R-
modules. Let M be a right R-module and S = EndR(M), its endomorphism ring. We
denote σ[M ] the full subcategory of Mod-R whose objects are submodules of M -generated
modules. A right R-module M is called a subgenerator, if it generates σ[M ] and a self-
generator, if it generates all its submodules. We denote the socle and radical of the right
R-module M by Soc(M) and Rad(M), respectively. The Jacobson radical of a ring R
is denoted by J(R). We use the notations l and r to denote left and right annihilator,
respectively.

The Baer Criterion has been generalized by many authors. For example, in 1989,
Camillo introduced the notion of principally injective modules for commutative rings
in [3]. A right R-module M is called principally injective (or p-injective), if every R-
homomorphism from a principal right ideal of R to M can be extended to one from R

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v11i3.3291

Email addresses: nghiemndh@gmail.com (N. D. H. Nghiem),
phatsarapa@gmail.com (P. Janmuang), samruam.b@chula.ac.th (S. Baupradist),
ronnason.c@psu.ac.th (R. Chinram)

http://www.ejpam.com 815 c© 2018 EJPAM All rights reserved.



N. D. H. Nghiem et al. / Eur. J. Pure Appl. Math, 11 (3) (2018), 815-822 816

to M . Next in 1999, Sanh and his group introduced M -principal injectivity for a given
right R-module M in [7]. Let M be a right R-module. A right R-module N is called
M -principally injective, if every R-homomorphism from an M -cyclic submodule of M to
N can be extended to one from M to N . Slightly compressible modules were studied
by Smith in [9]. In 2006, essentially compressible modules and rings were introduced
and studied by Smith and Vedadi in [10]. Essentially compressible modules is a one of
generalization of compressible modules. Next, the concepts of essentially slightly com-
pressible modules and rings were introduced, and related properties were investigated by
Singh in [8]. Essentially slightly compressible modules and rings are a generalization of
essentially compressible modules and rings. Celik introduced and investigated completely
slightly compressible modules as a one of generalization of compressible modules. Re-
cently, Baupradist et al. studied a general form of slightly compressible modules in [2].
That is, for a right R-module M and N , N is called an M -slightly compressible module,
if every non-zero submodule A of N , there exists a non-zero R-homomorphism from M
to A. In that paper, they provided conditions for right R-module to be an M -slightly
compressible module and examples of M -slightly compressible modules.

In this paper, we introduce the concept of M -slightly compressible-injective modules,
which extended from the Baer Criterion. Moreover, we study some properties of M -
slightly compressible-injective modules and relationship between M -principally injective
modules and M -slightly compressible-injective modules. For the some examples of M -
slightly compressible-injective modules are provided. For definitions and terminologies
not given in this paper, the reader is refereed to [1, 5, 6].

2. Slightly compressible injectivity

For a ring R, we see that every right ideal of R is an RR-slightly compressible sub-
module of RR and every RR-slightly compressible submodule of RR is a right ideal of
R because every submodule of RR is a right ideal of R. We use this fact to generalize
the notion of injectivity to an M -slightly compressible-injective module for a given right
R-module M . By an M -cyclic submodule, we mean the submodule of M of the form s(M)
with s ∈ S = EndR(M).

Definition 1. ([2]) Let M be a right R-module. A submodule A of M is called an M -
slightly compressible submodule of M , if every non-zero submodule A of N , there exists a
non-zero R-homomorphism s from M to N such that s(M) ↪→ A. A right R-module N is
called quasi-slightly compressible, if N is an N -slight compressible module.

Definition 2. Let M be a right R-module. A right R-module N is called an M -
slightly compressible-injective module (or M -sc-injective module for short), if every R-
homomorphism from an M -slightly compressible submodule of M to N can be extended
to an R-homomorphism from M to N . A right R-module N is called quasi-slightly
compressible-injective (or quasi-sc-injective for short), if N is an N -slightly compressible-
injective module.



N. D. H. Nghiem et al. / Eur. J. Pure Appl. Math, 11 (3) (2018), 815-822 817

Example 1.
(1) Every simple right R-module is a quasi-slightly compressible-injective module.

(2) The following example [see [5], Exercise(2), p.361], let F be a field andR=

(
F F
0 F

)
be the ring of all matrices of the form

(
a b
0 c

)
with a, b, c ∈ F . Let M =

(
F F
0 0

)
be a

right R-module of all matrices of the form

(
a b
0 0

)
with a, b ∈ F and N =

(
0 0
0 F

)
be a

right R-module of all matrices of the form

(
0 0
0 c

)
with c ∈ F . Then N is an M -slightly

compressible-injective module.

Proof. Let A be a non-zero M -slightly compressible submodule of M and α an R-

homomorphism from A to N . Then A has the form

(
0 F1

0 0

)
,

(
F2 0
0 0

)
or

(
F3 F4

0 0

)
where F1, F2, F3 and F4 are subfields of F .

Let α ∈ HomR(A,N) such that any element x ∈ A, α(x) =

(
0 0
0 αx

)
∈ N. It is easy

to define α from M to N by α(

(
a b
0 0

)
) =

(
0 0
0 αx

)
. It is clear that ᾱ|A = α. Therefore

N is an M -slightly compressible-injective module.

Proposition 1. Let M be a right R-module and A be a non-zero submodule of M . If A
is an M -slightly compressible-injective module, then A is a direct summand of M .

Proof. Assume that A is an M -slightly compressible-injective module. Then there
exists α : M → A such that αiA = IA where iA is the inclusion map from A to M and IA
is the identity map on A. So A is a direct summand of M .

Proposition 2. Let M be a quasi-slightly compressible-injective module and f, g ∈ S =
EndR(M). Then f ∈ Sg if and only if Ker(g) ⊆ Ker(f).

Proof. (⇒) Obviously.
(⇐) Assume that Ker(g) ⊆ Ker(f). By the Factor’s Theorem, there exists g′ :

g(M)→M such that g′g = f . Since M is a quasi-slightly compressible-injective module,
there exists h ∈ S such that hig(M) = g′ where ig(M) : g(M) → M is an embedding. So
hg = hig(M)g = g′g = f . Therefore f ∈ Sg.

Proposition 3. Let M and N be right R-modules. If N is an M -slightly compressible-
injective module, then any R-monomorphism from N to M splits.

Proof. Assume that N is an M -slightly compressible-injective module. Let f :
N → M be an R-monomorphism. Thus f−1 : f(N) → M is well defined and is



N. D. H. Nghiem et al. / Eur. J. Pure Appl. Math, 11 (3) (2018), 815-822 818

an R-homomorphism. Since f(N) is an M -slightly compressible submodule of M , f−1

can be extended to an R-homomorphism α : M → N such that αif(N) = f−1 where
if(N) : f(N) → M is an embedding. Therefore αf = IN where IN is an identity map on
N and hence f splits.

Proposition 4. Let M and N be right R-modules. If N is an M -slightly compressible
injective module and A ⊂⊕> N , then A is an M -slightly compressible injective module.

Proof. Assume that N is an M -slightly compressible injective module and A ⊂⊕> N .
Let B be a non-zero M -slightly compressible submodule of M and α : B → A be an
R-homomorphism. Since A ⊂⊕> N , there exists A′ ↪→ N such that N = A ⊕ A′. Let
iA : A → N be the canonical injection map. Since N is an M -slightly compressible
injective module, there exists f : M → N such that fiB = iAα where iB : B → M is an
embedding. Let πA : N → A be the canonical projection map. We can choose ᾱ = πAf .
Then ᾱiB = πAfiB = πAiAα = IAα = α where IA is the identity on A. Hence A is an
M -slightly compressible-injective module.

Proposition 5. LetM be a right R-module. IfM is a quasi-slightly compressible-injective
module, then every submodule of M which is isomorphic to a direct summand of M is a
direct summand of M.

Proof. Let M be a quasi-slightly compressible-injective module, A ⊂⊕> M and B ↪→M
such that A ∼= B. By Proposition 4, A is an M -slightly compressible-injective module.
Since A ∼= B, B is an M -slightly compressible-injective module. By Proposition 3, we
have iB : B → M splits where iB is a monomorphism from B to M . Therefore B is a
direct summand of M .

Proposition 6. Let M , N be right R-modules and N be an M -slightly compressible-
injective module. Then

(1) N is a K-slightly compressible-injective module for all K ⊂⊕> M .

(2) H is a K-slightly compressible-injective module for all H ⊂⊕> N and K ⊂⊕> M .

Proof. (1) Let K ⊂⊕> M and 0 6= A be an K-slightly compressible submodule and
α be an R-homomorphism from A to N . Then there exists 0 6= s ∈ EndR(K) such
that s(M) ↪→ A, so sπK ∈ EndR(M) and sπK(M) ↪→ A where πK : M → K is the
canonical map. Since N is an M -slightly compressible-injective module, α extends to an
R-homomorphism ᾱ from M to N such that ᾱiA = α where iA : A→M is an embedding.
Thus ᾱ|K : K → N and ᾱ|KiA = α. Therefore N is an K-slightly compressible-injective
module.

(2) Let H ⊂⊕> N and K ⊂⊕> M . From (1), N is K-slightly compressible injective. By
Proposition 4, H is an K-slightly compressible injective module.

Recall that a right R-module M is said to be direct-projective, if given any summand
N of M with projection map p : M → N and any epimorphism f : M → N , there exists



N. D. H. Nghiem et al. / Eur. J. Pure Appl. Math, 11 (3) (2018), 815-822 819

g ∈ S = EndR(M) such that fg = p. For more details of direct-projective, we refer to
[12].

Theorem 1. Let M be a right R-module and S = EndR(M) be the endomorphism ring of
M . If M is a direct-projective and every submodule of M is an M -slightly compressible-
injective module, then S is a von Neumann regular.

Proof. Assume that M is a direct-projective and every submodule of M is an M -
slightly compressible injective module. Let s ∈ S. By assumption, s(M) is an M -slightly
compressible injective module. Let is(M) : s(M) → M be an embedding. By Proposition
3, is(M) : s(M) → M splits. Then s(M) is a direct summand of M . We can construct
epimorphism s′ : M → Im(s) by s′(m) = s(m) for all m ∈ M . Since M is a direct-

projective, then the short exact sequence 0 → Ker(s′) ↪→ M
s′→ s(M) → 0 split and

we have Ker(s′) is a direct summand M. But Ker(s′) = Ker(s). So Ker(s) is a direct
summand of M . From Proposition 37.7(1) in [11], there exists g ∈ S such that s = sgs.
Therefore S is a von Neumann regular.

Theorem 2. Let M be a right R-module and S = EndR(M) be the endomorphism ring
of M .

(1) If M is a quasi-slightly compressible-injective module, then lS(Ker(s)) = Ss for all
s ∈ S.

(2) If M is a quasi-slightly compressible-injective module, then Ker(t) ⊆ Ker(s) implies
Ss ⊆ St for any s, t ∈ S.

(3) If M is a quasi-slightly compressible-injective module, then lS(Im(t) ∩ Ker(s)) =
lS(Im(t)) + Ss for all s, t ∈ S.

Proof. (1) Assume that M is a quasi-slightly compressible-injective module. It is easy
to show that Ss ⊆ lS(Ker(s)). Let s ∈ S and u ∈ lS(Ker(s)). We have u(Ker(s)) = 0.
Then Ker(s) ⊆ Ker(u). By the Factor’s Theorem, there exists an R-homomorphism
α : s(M) → M such that αs = u. Since M is a quasi-slightly compressible-injective
module, there exists an R-homomorphism ᾱ : M → M such that ᾱ|s(M) = α. Then
u = αs = ᾱs ∈ Ss. Hence lS(Ker(s)) ⊆ Ss. Therefore lS(Ker(s)) = Ss.

(2) Assume that M is a quasi-slightly compressible-injective module. Let s, t ∈ S.
Suppose Ker(t) ⊆ Ker(s). By Proposition 2, s ∈ St. Therefore Ss ⊆ St.

(3) Assume that M is a quasi-slightly compressible-injective module. Let s, t ∈ S.
Suppose u ∈ lS(Im(t) ∩Ker(s)). Then u(Im(t) ∩Ker(s))) = 0 and we have Ker(st) ⊆
Ker(ut). By Factor’s Theorem, there exists a map g′ : st(M) → M such that g′st = ut.
SinceM is a quasi-slightly compressible-injective module, there exists anR-homomorphism
g : M → M such that g|st(M) = g′. Thus ut = gst. It follows that (u − gs)t = 0 and
hence u − gs ∈ lS(Im(t)). Thus u ∈ lS(Im(t)) + Ss. We have lS(Im(t) ∩ Ker(s)) ⊆
lS(Im(t)) + Ss. But it is clear that lS(Im(t)) + Ss ↪→ lS(Im(t) ∩ Ker(s)). Therefore
lS(Im(t) ∩Ker(s)) = lS(Im(t)) + Ss for all s, t ∈ S.



N. D. H. Nghiem et al. / Eur. J. Pure Appl. Math, 11 (3) (2018), 815-822 820

Theorem 3. Let M be a quasi-slightly compressible module, S = EndR(M), ∆ be the set
of all s ∈ S such that Ker(s) is an essential in M and J(S) be the Jacobson radical of S.
If M is a quasi-slightly compressible-injective module and every M -cyclic submodule of M
is an injective, then J(S) = ∆.

Proof. Assume that M is a quasi-slightly compressible-injective module and every
M -cyclic submodule of M is an injective. Let s ∈ ∆. Then Ker(s) is an essential in M .
Since Ker(s) ∩Ker(1 − s) = 0, Ker(1 − s) = 0, lS(Ker(1 − s)) = S. By Theorem 2(2),
lS(Ker(1 − s)) = S(1 − s). Then S(1 − s) = S. Hence 1 − s has left inverse in S. By
Theorem 9.3.1 in [5], ∆ ⊆ J(S). Next, let s ∈ J(S). We want to show that Ker(s) is an
essential in M . First, we claim that if Im(t) ∩Ker(s) = 0 for all t ∈ S, then t = 0. Let
t ∈ S such that Im(t)∩Ker(s) = 0. By Theorem 2(3), lS(Im(t)∩Ker(s)) = lS(Im(t))+Ss
but lS(Im(t)∩Ker(s)) = S. Then lS(Im(t))+Ss = S. Since s ∈ J(S), Ss is a small in S,
lS(Im(t)) = S, Im(t) = 0. Then t = 0. Let A ↪→M such that Ker(s)∩A = 0. Since M is
a quasi-slightly compressible module and every M -cyclic submodule is an injective, from
Corollary 2.14 in [2], M is a self-generator, A =

∑
t∈I t(M) where I ⊆ S = EndR(M),∑

t∈I t(M) ∩Ker(s) = 0, t(M) ∩Ker(s) = 0 for all t ∈ I. We have t = 0 for all t ∈ I.
Then A =

∑
t∈I t(M) = 0, so Ker(s) is an essential in M . Hence s ∈ ∆, J(S) ⊆ ∆.

Therefore J(S) = ∆.

Theorem 4. Let M be a quasi-slightly compressible-injective module and s, t ∈ S =
EndR(M). If s(M) ∼= t(M), then Ss ∼= St.

Proof. Assume that s(M) ∼= t(M). Then there exists an isomorphism f from s(M) to
t(M). Since M is a quasi-slightly compressible-injective module, s(M) is an M -slightly
compressible submodule, it(M)f : s(M) → M is an R-homomorphism, so it(M)f can be
extended to f̄ : M → M such that f̄ is(M) = it(M)f where is(M) : s(M) → M and
it(M) : t(M) → M are embedding. Define β : St → Ss by β(ut) = uf̄s for all u ∈
S. Since Im(f̄ s) = Im(t), we can show that β is an well-defined. Moreover, β is a
left S-homomorphism. For any v ∈ S, vis(M) : s(M) → M can be extended to an
R-homomorphism ϕ : M → M such that ϕit(M)f = vis(M) and we can construct the
map s′ : M → s(M) such that s′(m) = s(m) for all m ∈ M , so is(M)s

′ = s where
is(M) : s(M) → M and it(M) : t(M) → M are embedding. We have β(ϕt) = ϕf̄s =
ϕf̄is(M)s

′ = ϕit(M)fs
′ = vis(M)s

′ = vs. This shows that β is an epimorphism. It is clear
that β is a left S-monomorphism. Therefore Ss ∼= St.

Theorem 5. Let M be a quasi-slightly compressible-injective module and s1, . . . , sn ∈
S = EndR(M) such that the sum

∑n
i=1 Ssi is direct. Then any R-homomorphism from∑n

i=1 si(M) to M can be extended to an R-homomorphism from M to M .

Proof. Since (
∑n

i=1 si) (M) ⊆
∑n

i=1 si(M) and M is a quasi-slightly compressible-
injective module, so any R-homomorphism from

∑n
i=1 si(M) to M can be extended to an

R-homomorphism from M to M .



N. D. H. Nghiem et al. / Eur. J. Pure Appl. Math, 11 (3) (2018), 815-822 821

Theorem 6. Let M be a quasi-slightly compressible-injective module, s1, . . . , sn ∈ S =
EndR(M) such that the sum

∑n
i=1 Ssi is direct, A = s1(M) + . . . + sk(M) and B =

sk+1(M) + . . .+ sn(M) where 1 ≤ k ≤ n. Then lS(A ∩B) = lS(A) + lS(B).

Proof. Clearly, lS(A ∩ B) ⊇ lS(A) + lS(B). Let u ∈ lS(A ∩ B). Consider the map
α : A + B → M by α(a + b) = u(a) for all a ∈ A, b ∈ B. Since u(A ∩ B) = 0, α is
well-defined and is an R-homomorphism. By Theorem 5, α : A+B →M can be extended
to an R-homomorphism ϕ : M →M . Clearly, ϕ(b) = 0 for all b ∈ B and hence ϕ ∈ lS(B)
and u− ϕ ∈ lS(A). Therefore u = (u− ϕ) + ϕ ∈ lS(A) + lS(B).

Theorem 7. Let M and Mi be right R-modules for all i ∈ I = {1, 2, ..., n} where n is a
positive integer. Then Mi is an M -slightly compressible-injective module for all i ∈ I if
and only if ⊕ni=1Mi is an M -slightly compressible-injective module.

Proof. (⇒) Assume that Mi is an M -slightly compressible-injective module for all
i ∈ I. Let j ∈ I, P = ⊕ni=1Mi, A be a non-zero M -slightly compressible submodule of M
and α be an R-homomorphism from A to P . Since πiα is an R-homomorphism from A
to Mi where πj the jth canonical projection map from P to Mj and Mj is an M -slightly
compressible-injective module, there exists ᾱj : M → Mj such that ᾱjiA = πjα where
iA : A → M is an embedding. We can choose ᾱ =

∑n
j=1 ijᾱj where ij : Mj → M be

the canonical injection map. Then ᾱiA =
∑n

j=1 ijᾱjiA = (
∑n

j=1 ijπj)α = IPα = α where
IP is the identity map on P . Hence P = ⊕ni=1Mi is an M -sligthly compressible-injective
module.

(⇐) Assume that P = ⊕ni=1Mi is an M -slightly compressible-injective module. Let
j ∈ I, A be a non-zero M -slightly compressible submodule of M and αj be an R-
homomorphism from A to Mj . Since ijαj is an R-homomorphism from A to P where
ij : Mj →M is the canonical injection map and P is an M -slightly compressible-injective
module, there exists ᾱ : M → P such that ᾱiA = ijαj where iA : A → M is an em-
bedding. We can choose ᾱj = πjᾱ where πj is the jth canonical projection map. Then
ᾱjiA = πjᾱiA = πjijαj = IMjαj = αj where IMj is an identity map on Mj . Hence Mi is
an M -slightly compressible-injective module for all i ∈ I.

Theorem 8. Let M be a quasi-slightly compressible module. Then M is a semisimple
module if and only if every non-zero submodule of M is M -slightly compressible-injective.

Proof. (⇒) It is easy.
(⇐) Assume that every non-zero submodule of M is an M -slightly compressible-

injective module. Let A be a submodule of M . If A = 0, then we are done. Suppose that
0 6= A ↪→M . By assumption, A is an M -slightly compressible-injective module. Since M
is a quasi-slightly compressible module, A is an M -slightly compressible submodule of M .
Then there exists α : M → A such that αiA = IA where iA : A→M is the embedding and

IA is an identity map of A. Then the short exact sequence 0 → A
iA−→ M

π−→ M/A → 0
splits. Thus A is a direct summand of M . Therefore M is a semisimple module.



REFERENCES 822

Acknowledgements

The first author was supported from Rachadapisek Sompot Fund for Postdoctoral
Fellowship, Chulalongkorn University.

We would like to thank the reviewers of this paper for their time to read our manuscript
carefully and their comments and suggestions.

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