Vol 8, No 1 (2017)

Available online throug

ISSN 2229 – 5046

NONSINGULAR PQ-INJECTIVE MODULES

S. WONGWAI*, N. PARINYUD AND C. KHAMPARAT

Faculty of Architecture,

Rajamangala University of Technology Thanyaburi, Pathumthani 12110, Thailand.

(Received On: 11-11-16; Revised & Accepted On: 09-01-17)

ABSTRACT

L

et

M

be a right

R

-module. A right

R

-module

N

is called nonsingular principally

M

-injective (briefly, nonsingular PM-injective) if, for each m∈M \ Z(M),any

R

-homomorphism from mRto

N

can be extended to an

R

-homomorphism from

M

to

N

.

M

is called nonsingular principally quasi –injective (briefly, nonsingular

PQ

-injective) if, it is nonsingular

PM

-injective. In this paper, we give some characterizations and properties of nonsingular

PQ

-injective modules.

2010 Mathematics Subject Classification: 13C10, 13C11, 13C60.

Key words and phrases: Nonsingular _PQ-injective Modules and Endomorphism Rings.

1. INTRODUCTION

Let

R

be a ring. 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 an

R

-homomorphism from

R

to

M.

Equivalently,

l r (a)

_{M R}

=

Ma

for all

a

∈

R

where

l

and

r

are left and right annihilators, respectively. This notion was introduced by Camillo [2] for commutative rings. In [8], Nicholson and Yousif studied the structure of principally injective rings and gave some applications. Nicholson, Park, and Yousif [9] extended this notion of principally injective rings to the one for modules. In [5], W. Junchao introduced the definition of

Jcp

-injective rings, a ring

R

is called right

Jcp

-injective if for each

a

∈

R Z

\

_r

,

any

R

-homomorphism from

aR

to

R

can be extended to an

R

-homomorphism from

R

to

R.

A right

R

-module

M

is called almost mininjective [11] if, for any simple right ideal

kR

of

R

, there exists an

S

-submodule

X

_k of

M

such that

M R k

l (r (m))=Mk⊕X as left

S

-modules. A ring

R

is called right almost mininjective if

R

_R is almost mininjective. In this note we introduce the definition of nonsingular

PQ

-injective modules and give some characterizations and properties.

Throughout this paper,

R

will be an associative ring with identity and all modules are unitary right

R

−

modules. For right

R

−

modules

M

and

N,

Hom (M, N)

_R denotes the set of all

R

−

homomorphisms from

M

to

N

and

R

S

=

End (M)

denotes the endomorphism ring of

M.

If

X

is a subset of

M

the right (resp. left) annihilator of

X

in

R

(resp.

S

) is denoted by

r (X)

_R (resp.

l (X)

_S ). By notation

N

⊂

⊕

M

(N

⊂

e

M)

we mean that

N

is a direct summand (an essential submodule) of

M.

We denote the singular submodule of

M

by

Z(M).

2. NONSINGULAR

PM

-INJECTIVE MODULES

Recall that a submodule

K

of a right

R

−

module

M

is essential (or large) in

M

if, every nonzero submodule

L

of

M

, we have

K

∩ ≠

L

0.

An element

m

∈

M

is called singular if

r (m)

_R

⊂

e

R.

M

is called nonsingular if it contains no nontrivial singular element.

Corresponding Author: S. Wongwai*, Faculty of Architecture,

Definition 2.1: Let

M

be a right

R

-module. A right

R

-module

N

is called nonsingular principally

M

-injective (briefly, nonsingular

PM

-injective) if, for each

m

∈

M \ Z(M),

any

R

-homomorphism from

mR

to

N

can be extended to an

R

-homomorphism from

M

to

N.

Lemma 2.2: Let

M

and

N

be right

R

−

modules. Then

N

isnonsingular

PM

-injective if and only if for each

m

∈

M \ Z(M),

R N R

Hom (M, N)m

=

l r (m).

Proof: Clearly,

Hom (M, N)m

_R

⊂

l r (m).

_{N R}

Let

x

∈

l r (m).

_{N R} Define

ϕ

: mR

→

xR

by

ϕ

(mr)

=

xr

for every

r

∈

R.

Since

r (m)

_R

⊂

r (x),

_R

ϕ

is well-defined. It is clear that

ϕ

is an

R

−

homomorphism. Since

N

is nonsingular

PM

-injective, there exists an

R

−

homomorphism

ϕ



: M

→

N

such that

ϕι = ι ϕ



_{1 2}

,

where

ι

₁

: mR

→

M

and

ι

₂

: xR

→

N

are the inclusion maps. Hence

x

= ϕ

(m)

= ϕ



(m)

∈

Hom (M, N)m.

_R

Conversely, let

m

∈

M \ Z(M),

and

ϕ

: mR

→

N

be an

R

−

homomorphism. Then

ϕ

(m)

∈

l r (m)

_{N R} so by assumption, we have

ϕ

(m)

= ϕ



(m)

for some

ϕ∈



Hom (M, N).

_R This shows that

N

is nonsingular

PM

-injective.

Example 2.3: Let

R

F

F

0

F





= 







where

F

is a field.

(1) If

M

_R

F

F

0

0





= 







and R

0

0

N

0

F





= 







, then

N

is not nonsingular

PM

-injective.

(2) If

M

_R

0

F

0

F





= 







and R

F

F

N

0

0





= 







, then

N

is nonsingular

PM

-injective.

Proof: (1) It is clear that only

0

0

,

0

F













0

F

0

0













and

F

F

0

0













are nonzero nonessential principal right ideals of

R.

Let

m

x

y

M

0

0





=

_

_

∈





with

x

≠

0

or

y

≠

0.

Then

m

∈

M \ Z(M)

and that nonzero submodules

mR

of

M

may be

0

F

0

0













or

M.

It is clear that

0

F

0

0

.

0

0

0

F



 





 





 





For any

R

−

homomorphism

F

F

0

0

:

0

0

0

F









ϕ

_

_

→

_

_









with

1

0

0

0

0

0

0

x













ϕ

_

_

_

_

=

_

_













for some

x

∈

F

,

a

b

1

0

a

b

0

0

a

b

0

0

0

0

0

0

0

0

0

x

0

0

0

0







 













 



ϕ

_

_

_

_{ }

=

_

_

_

_

=

_

_

_{ }

=

_















 





 



for every

a

b

F

F

0

0

0

0



 



∈



 





 



, hence

ϕ =

0.

Then

N

is not nonsingular

PM

-injective.

(2) For

M

_R

0

F

0

F





= 







and R

F

F

N

0

0





= 







, let

0

x

m

M

0

y





=

_

_

∈





where

x

≠

0

or

y

≠

0.

Then

R

F

F

r (m)

0

0





= 







is a nonessential right ideal of

R

and

mR

may be

0

F

,

0

0













0

0

0

F













or

M.

Let

:

0

F

F

F

0

0

0

0









α

_

_

→

_

_

Then there exists

x , x

_{1 2}

∈

F

such that

0

1

x

1

x

2

0

0

0

0













α

_

_

_

_

=

_

_













.

Hence

0

1

0

1

0

0

0

0

0

0

0

1



















α

_

_

_

_

= α

_

_

_

_

_



















0

1

0

0

x

1

x

2

0

0

0

x

2

0

0

0

1

0

0

0

1

0

0





 



 



 



= α

_

_

_{ }

_

_{ }

=

_

_{ }

=

_



 

 



 







.

It follows that

x

₁

=

0

.

Define

α



: M

→

N

by



0

a

0

ax

2

0

b

0

0













α

_

_

_

_

=

_

_













for every

a, b

∈

F

.

It is clear that

α



is an

R

−

homomorphism and



0

1



0

1

1

0

0

x

2

1

0

0

x

2

.

0

0

0

0

0

1

0

0

0

1

0

0













 



 



 



α

_

_

_

_

= α

_

_

_{ }

_

_{ }

=

_

_{ }

=

_







 

 



 











This shows that



α

is an extension of

α

.

Then

N

is nonsingular

PM

-injective.

Lemma 2.4:

(1)

N

is nonsingular

PM

-injective if and only if

N

is nonsingular

PX

-injective for any submodule

X

of

M.

(2)

⊕

n_{i 1=}

N

_i isnonsingular

PM

-injective if and only if

N

_i is nonsingular

PM

- injective for all

i.

(3) If

m

∈

M \ Z(M)

and

mR

is nonsingular

PM

-injective, then

mR

⊂

⊕

M.

Proof:

(1) The sufficiency is trivial. For the necessity, let

x

∈

X \ Z(X),

and

ϕ

: xR

→

N

be an

R

−

homomorphism. Since

x

∈

M \ Z(M),

there exists an

R

−

homomorphism

ϕ



: M

→

N

such that

ϕ = ϕι ι



_{2 1} where

1

: xR

X

ι

→

and

ι

₂

: X

→

M

are the inclusion maps. Then

ϕι



₂ extends

ϕ

.

(2) The necessity is trivial. For the sufficiency, let

m

∈

M \ Z(M),

and

ϕ

: mR

→ ⊕

n_{i 1=}

N

_i be an

R

−

homomorphism. Then for each

i

, there exists

R

−

homomorphisms

ϕ

_i

: M

→

N

_i such that

ϕ ι = π ϕ

_{i i} where

π ⊕

_i

:

_{i 1}n₌

N

_i

→

N

_i is the projection map, and

ι

: mR

→

M

is the inclusion map. Put

n

1 1 n n i 1 i

ˆ

...

: M

₌

N .

ϕ = ι ϕ + + ι ϕ

→ ⊕

Then it is clear that

ϕ

ˆ

extends

ϕ

.

(3) Since

mR

is nonsingular

PM

-injective, there exists an

R

−

homomorphism

ϕ

: M

→

mR

such that

mR

1

ϕι =

where

ι

: mR

→

M

is the inclusion map. Then by[1, Lemma 5.1] ,

ι

is a split monomorphism, therefore

mR

⊂

⊕

M.

Theorem 2.5: The following conditions are equivalent for a projective module

M.

(1) Every

m

∈

M \ Z(M),

mR

is projective.

(2) Every factor module of a nonsingular

PM

-injective module is nonsingular

PM

- injective. (3) Every factor module of an injective

R

−

module is nonsingular

PM

-injective.

Proof:

(1)

⇒

(2) :

Let

N

be a nonsingular

PM

-injective module,

X

a submodule of

N

,

m

∈

M \ Z(M),

and

: mR

N / X

(2)

⇒

(3) :

is clear.

(3)

⇒

(1) :

Let

m

∈

M \ Z(M),

h : A

→

B

an

R

−

epimorphism, and let

α

: mR

→

B

be an

R

−

homomorphism. Embed

A

in an injective module

E

[1, 18.6]. Let

σ

: B

→

A / Ker(h)

be an

R

−

isomorphism. Since

E / Ker(h)

is nonsingular

PM

-injective, there exists an

R

−

homomorphism



_{: M}

_{E / Ker(h)}

α

→

such that

ι σα = αι

₁



₂ where

ι

₁

: A / Ker(h)

→

E / Ker(h)

and

ι

₂

: mR

→

M

are the

inclusion maps. Since

M

is projective,

α



can be lifted to

β

: M

→

E.

Let

x

∈

mR.

Then

(x)

a

Ker(h)

σα

= +

for some

a

∈

A,

so

β

(x) Ker(h)

+

= ηβ

(x)

= α



(x)

= σα

(x)

= +

a

Ker(h)

where

: E

E / Ker(h)

η

→

is the natural

R

−

epimorphism. Hence

β

(x) a

− ∈

Ker(h)

⊂

A

so

β

(x)

∈

A.

This shows that

β

(mR)

⊂

A.

Therefore we have lifted

α

.

3. NONSINGULAR

PQ

-INJECTIVE MODULES

A right

R

-module

M

is called nonsingularprincipally quasi –injective (briefly, nonsingular

PQ

-injective) if, it is nonsingular

PM

-injective.

Lemma 3.1: Let

M

be a right

R

−

module and

S

=

End (M).

_R Then the following conditions are equivalent. (1)

M

is nonsingular

PQ

-injective.

(2)

l r (m)

_{M R}

=

Sm

for each

m

∈

M \ Z(M).

(3)

r (m)

_R

⊂

r (n),

_R where

m, n

∈

M

with

m

∈

M \ Z(M),

implies that

Sn

⊂

Sm.

(4)

l (r (m)

_{M R}

∩

aR)

=

l (a) Sm

_M

+

for all

a

∈

R

and

m

∈

M

with

ma

∈

M \ Z(M).

(5) If

α

: mR

→

M

is an

R

−

homomorphism,

m

∈

M \ Z(M),

then

α

(m)

∈

Sm.

Proof:

(1)

⇔

(2) :

by Lemma 2.2

(2)

⇒

(3) :

If

r (m)

_R

⊂

r (n),

_R where

m, n

∈

M

with

m

∈

M \ Z(M),

then

l r (n)

_{M R}

⊂

l r (m).

_{M R} Then

M R M R

Sn

⊂

l r (n)

⊂

l r (m)

=

Sm

by (2).

(3)

⇒

(4) :

Let

a

∈

R

and

m

∈

M

with

ma

∈

M \ Z(M)

and let

x

∈

l (r (m)

_{M R}

∩

aR).

Then

R R

r (ma)⊂r (xa),

and hence by (3),

Sxa

⊂

Sma.

Thus

xa

= ϕ

(ma),

ϕ∈

S

and so

(x

− ϕ

(m))

∈

l (a).

_M It follows that

M

x

∈

l (a) Sm.

+

The other hand is clear.

(4)

⇒

(5) :

Put

a

=

1

_R in (4), then

α

(m)

∈

l r (m)

_{M R}

=

l (r (m)

_{R R}

∩

1R)

=

l (1 ) Sm

_{M R}

+

=

Sm

because

m1 M \ Z(M).

∈

(5)

⇒

(1) :

Let

m

∈

M

with

m

∈

M \ Z(M)

and let

ϕ

: mR

→

M

be an

R

−

homomorphism. Then by (5),

(m)

Sm

ϕ

∈

so there exists an

R

−

homomorphism

ϕ∈

ˆ

S

is an extension of

ϕ

to

M.

Theorem 3.2: Let

M

be a nonsingular

PQ

-injective module and

m, n

∈

M

with

m

∈

M \ Z(M).

(1) If

mR

embeds into

nR,

then

Sm

is an image of

Sn.

(2) If

nR

is an image of

mR,

then

Sn

can be embedded into

Sm.

(3) If

mR



nR,

then

Sm



Sn.

Proof:

(1) Let

σ

: mR

→

nR

be an

R

−

monomorphism and let

ι

₁

: mR

→

M

and

ι

₂

: nR

→

M

be the inclusion maps. Since

M

is nonsingular

PQ

-injective, there exists an

R

−

homomorphism

σ



: M

→

M

such that

σι = ι σ



_{1 2}

.

Let

ϕ

: Sn

→

Sm

defined by

ϕ α

( (n))

= ασ



(m)

for every

α ∈

S

. Since



( (n))

( (m))

( (m))

(nR),

Since

σ

is monic,

r ( (m))

_R

σ

⊂

r (m)

_R and

σ

(m)

∈

M \ Z(M)

and hence by Lemma 3.1,

Sm

⊂ σ

S (m).

Then

m

∈ σ

S (m)

⊂ ϕ

(Sn).

(2) By the same notations as in (1), let

σ

: mR

→

nR

be an

R

−

epimorphism. Write

σ

(ms)

=

n,

s

∈

R.

Since

M

is nonsingular

PQ

-injective,

σ

can be extended to

σ



: M

→

M

such that

σι = ι σ



_{1 2}

.

Define

: Sn

Sm

ϕ

→

by

ϕ α

( (n))

= ασ



(ms)

for every

α ∈

S

. It is clear that

ϕ

is an

S

−

homomorphism. If

(n)

Ker( )

α

∈

ϕ

, then

0

= ϕ α

( (n))

= ασ



(ms)

= α

(n).

This shows that

ϕ

is an

S

−

monomorphism. (3) Follows from (1) and (2).

Recall that a right

R

-module

M

is called

C2

[6] if, every submodule of

M

that is isomorphic to a direct summand of

M

is itself a direct summand of

M.

M

is called

C3

if, whenever

N

and

K

are direct summands of

M

with

N

∩ =

K

0

then

N

⊕

K

also a direct summand of

M.

Theorem 3.3: Let

M

=

mR

be a principle, nonsingular

PQ

-injective module.

(1) If

X



e(mR),

e

2

= ∈

e

S

and

e(m)

∈

M \ Z(M),

then

X

=

g(mR),

for some

g

2

= ∈

g

S.

(2) If

e(mR)

∩

f (mR)

=

0,

e

2

= ∈

e

S, f

2

= ∈

f

S

and

f (m)

∈

M \ Z(M),

then

e(M)

⊕

f (M)

=

g(M),

for some

g

2

= ∈

g

S.

Proof:

(1) Let σ: e(mR)→X be an

R

−

isomorphism. Write

σ

e(m)

=

x

where

x

∈

X

so xR=X. We must show that

xR

=

g(mR),

for some

g

2

= ∈

g

S.

Then by Lemma 2.3, we have e(mR) is nonsingular

PM

-injective and hence

xR

is also nonsingular

PM

-injective. Since

σ

e(m)

∈

M \ Z(M),

xR

⊂

⊕

mR

Lemma 2.4.

(2) Let

e(mR)

∩

f (mR)

=

0,

2 2

e = ∈e S, f = ∈f S and

f (m)

∈

M \ Z(M).

Then

e(mR)

⊕

f (mR)

=

e(mR)

⊕ −

(1 e)f (mR).

If

(1 e)f (mR)

−

=

0,

then

e(mR)

⊕

f (mR)

is a direct summand of

M.

If (1 e)f (mR)− ≠0, then (1 e)f (mR)− f (mR), and hence

(1 e)f (mR)

−

=

g(mR)

for some 2

g = ∈g S by (1). Let

h

= + −

e g ge,

then

h

2

=

h

and e(M)⊕f (M)=h(M).

Lemma 3.4: Let

M

be a nonsingular

PQ

-injective module and

S

=

End (M).

_R If

α ∈

S

with

α

(M)

⊂

M \ Z(M),

then

l (Ker( )

_S

α ∩

mR)

=

l (m) S .

_S

+ α

Proof: Clearly,

S S

l (m) S+ α ⊂l (Ker( )α ∩mR). Let

S

l (Ker( ) mR).

β∈ α ∩ Then r ( (m))_R α ⊂r ( (m)),_R β so

M R M R

l r ( (m))

β

⊂

l r ( (m)).

α

Since

α

(m)

∈

M \ Z(M),

S (m)

β

⊂

l r ( (m))

_{M R}

β

⊂

l r ( (m))

_{M R}

α

= α

S (m)

by Lemma 3.1, so

β

(m)

= α

s (m)

for some

s

∈

S.

It follows that

(

β − α ∈

s )

l (m),

_S and hence

β∈

l (m) S .

_S

+ α

Following [8], a right

R

-module

M

is called a principal self-generator, if every element

m

∈

M

has the form

1

m= γ(m ) for some γ: M→mR. If

uR

≠

0

is uniform, we call

u

a uniform element of

M.

We call a right

R

-module

M

is a duo module if every submodule of

M

is fully invariant.

Theorem 3.5: Let

M

be a principal module which is a principal self-generator. Then the following conditions are equivalent.

(1)

M

is nonsingular

PQ

-injective.

(2)

l (Ker( )

_S

α ∩

mR)

=

l (m) S

_S

+ α

for all

m

∈

M

and

α ∈

S

with

α

(M)

∈

M \ Z(M).

(3)

l (Ker( ))

_S

α = α

S

for all

α ∈

S

with

α

(M)

∈

M \ Z(M).

(4)

Ker( )

α ⊂

Ker( ),

β

where

α β∈

,

S

with

α

(m)

∈

M \ Z(M),

implies that

S

β ⊂ α

S .

Proof:

(1)

⇒

(2) :

by Lemma 3.4.

(3)

⇒

(4) :

Ker( )

α ⊂

Ker( ),

β

then

l (Ker( ))

_S

β ⊂

l (Ker( )).

_S

α

It follows that

S

β ⊂

l (Ker( ))

_S

β ⊂

l (Ker( ))

_S

α = α

S .

(4)

⇒

(1) :

Let

m

∈

M \ Z(M),

ϕ

: mR

→

M

be an

R

−

homomorphism.

Since

M

is a principal self-generator, there exists

β∈

S

such that

β

(m )

₁

=

m,

so

Ker( )

β ⊂

Ker(

ϕβ

)

and

(M)

M \ Z(M).

β

⊂

Then by (4),

S

ϕβ ⊂ β

S

hence ϕβ = ϕβ for some

ϕ∈



S.

This shows that

ϕ



is an extension of

.

ϕ

Theorem 3.6: Let

M

be a duo, nonsingular

PQ

-injective module. If

u

a uniform element of

M

with

u

∈

M \ Z(M),

then

{

}

u

M = α ∈S Ker( )α ∩uR≠0 is a unique maximal left ideal of

S

containing

l (u).

_S

Proof: Since

uR

is uniform,

M

_u is a left ideal of

S.

It is clear that l (u)_S ⊂M_u ≠S. Let

X

be a left ideal of

S

containing

l (u)

_S and

X

≠

S.

If

α ∈ −

X

M ,

_u then Ker( )α ∩uR=0. Since

M

is a duo module,

(u)R

M \ Z(M)

α

⊂

and by Lemma 3.4 we have

S

=

l (Ker( )

_S

α ∩

uR)

=

l (u) S

_S

+ α ⊂

X

a contradiction. Thus

u

X

⊂

M .

Definition 3.7: Let

M

be a right

R

-module,

S

=

End (M)

_R . The module

M

is called almostnonsingular

PQ

-injective if, for each

m

∈

M \ Z(M),

there exists an

S

-submodule

X

_mof

M

such that

l (r (m))

_{M R}

=

Sm

⊕

X

_m as left

S

-modules.

Theorem 3.8: Let

M

be a right

R

-module,

S

=

End (M)

_R and

m

∈

M \ Z(M).

(1) If

Hom (mR, M)

_R

=

S

⊕

Y

as left

S

-modules, then

l (r (m))

_{M R}

=

Sm

⊕

X

as left

S

- modules, where

X

=

{

f (m) : f

∈

Y .

}

(2) If

l (r (m))

_{M R}

=

Sm

⊕

X

for some

X

⊂

M

as left

S

- modules, then we have

Hom (mR, M)

_R

= ⊕

S

Y

as left

S

- modules, where

Y

= ∈

{

f

Hom (mR, M) : f (m)

_R

∈

X .

}

(3)

Sm

is a direct summand of

l (r (m))

_{M R} as left

S

- modules if and only if

S

is a direct summand of

Hom (mR, M)

_R as left

S

- modules.

Proof: Define

θ

: Hom (mR, M)

_R

→

l (r (m))

_{M R} by

θ

(f )

=

f (m)

for every

f

∈

Hom (mR, M).

_R It is obvious that

θ

is an

S

-monomorphism. For

x

∈

l (r (m)),

_{M R} define

g : mR

→

M

by

g(mr)

=

xr

for every

r

∈

R.

Since

R R

r (m)⊂r (x),

g

is well-defined, so it is clear that

g

is an

R

-homomorphism. Then

θ

(g)

=

g(m)

=

x.

Therefore

θ

is an

S

-isomorphism. Let

α

(m)

∈

Sm.

Since

α

(m)

∈

l (r (m)),

_{M R} there exists

ϕ∈

Hom (mR, M)

_R such

that

θ ϕ = α

( )

(m),

so

ϕ

(m)

= α

(m).

Define ϕ: M→M by ϕ(x)= α(x) for every

x

∈

M.

It is clear that

ϕ



is

an

R

-homomorphism and is an extension of

ϕ

.

Then

α

(m)

= ϕ



(m)

= θ ϕ

( ).



This shows that

Sm

⊂ θ

(S).

The

other inclusion is clear. Then

θ

(S)

=

Sm

and X= θ(Y)=

{

f (m) : f∈Y .

}

Then the Lemma follows.

Theorem 3.9: The following conditions are equivalent: (1)

M

is almost nonsingular

PQ

-injective.

(2) There exists an indexed set

{

X : m

_m

∈

M

}

of

S

-submodules of

M

with the property that if mR⊂M \ Z(M),

m

∈

M,

then

l (r (m)

_{M R}

∩

aR)

=

(X

_ma

: a)

_l

+

Sm

and (X_ma: a)_l∩Sm⊂l (a)_M for all

a

∈

R,

where

(X

_ma

: a)

_l

=

{

n

∈

M : na

∈

X

_ma

}

if

ma

≠

0

and

ma l M

Proof:

(1)

⇒

(2)

Let m∈M \ Z(M). Then there exists an

S

-submodule

X

_m of

M

such that l (r (m))_{M R} =Sm⊕X_m as left

S

-modules. Let

a

∈

R.

If

ma

=

0,

then

R

aR⊂r (m) so (2) follows. If

ma

≠

0,

then any

M R

x∈l (r (m)∩aR)

we have

r (ma)

_R

⊂

r (xa)

_R and so

xa

∈

l (r (xa))

_{M R}

⊂

l (r (ma))

_{M R}

=

Sma

⊕

X

_mabecause

ma

∈

M \ Z(M).

Write

xa

= α

(ma)

+

y

where

α ∈

S

and

y

∈

X .

_ma Then (x− α(m))a= ∈y X ,_ma so

x

− α

(m)

∈

(X

_ma

: a) .

_l It follows that

x

∈

(X

_ma

: a)

_l

+

Sm.

This shows that l (r (m)_{M R} ∩aR)⊂(X_ma: a)_l+Sm. Conversely, it is clear that

Sm

⊂

l (r (m)

_{M R}

∩

aR).

Let

ma l

y

∈

(X

: a) .

Then

ya

∈

X

_ma

⊂

l (r (ma)).

_{M R} If

as

∈

r (m)

_R

∩

aR,

then

mas

=

0

and so

yas

=

0.

Hence

M R

y∈l (r (m)∩aR).This shows that

ma l

(X : a) ⊂l (r (m)_{M R} ∩aR). Therefore

l (r (m)

_{M R}

∩

aR)

=

(X_ma: a)_l+Sm. If

ma l

(m) (X : a) Sm,

β ∈ ∩ then β(m)a∈X_ma∩Sma=0. Hence

β

(m)

∈

l (a).

_M

(2)

⇒

(1)

Let

m

∈

M \ Z(M).

Then there exists an

S

-submodule

X

_m of

M

such that

M R M R m l

l (r (m))

=

l (r (m)

∩

R)

=

(X :1)

+

Sm

and (X :1)_{m l}∩Sm⊂l (1)_M =0. Note that (X :1)_{m l}=X ._m Then (1) follows.

ACKNOWLEDGMENT

The authors would like to thank the referees for their helpful comments. This research is supported by the Faculty of Architecture, Rajamangala University of Technology Thanyaburi.

REFERENCES

1. F. W. Anderson and K. R. Fuller, “Rings and Categories of Modules”, Graduate Texts in Math.No.13, Springer-verlag, New York, 1992.

2. Camillo V., Commutative rings whose principal ideals are annihilators, Portugal. Math., 46(1989), p. 33-37. 3. N. V. Dung, D. V. Huynh, P. F. Smith and R. Wisbauer, “Extending Modules”, Pitman, London, 1994. 4. Lam T. Y., A First Course in Noncommutative Rings, Graduate Texts in Mathematics Vol. 131,

Springer-Verlag, New York, 1991.

5. W. Junchao, Jcp-Injective Rings International Electronic Journal of Algebra, Volume 6 (2009) 1-22.

6. S. H. Mohamed and B. J. Muller, “Continuous and Discrete Modules”, London Math. Soc. Lecture Note Series 14, Cambridge Univ. Press, 1990.

7. W. K. Nicholson and M. F. Yousif, Principally injective rings, J. Algebra, 174(1995), 77--93.

8. W. K. Nicholson, J. K. Park and M. F. Yousif, Principally quasi-injective modules, Comm. Algebra, 27:4 (1999), 1683--1693.

9. R. Wisbauer, “Foundations of Module and Ring Theory”, Gordon and Breach London, Tokyo e.a., 1991. 10. S. Wongwai, On the endomorphism ring of a semi-injective module, Acta Math. Univ. Comenianae, Vol.71,

1(2002), pp. 27-33.

11. S. Wongwai, Almost Mininjective Rings, Thai Journal of Mathematics, 4(1) (2006), 245-249.

12. S. Wongwai, Small PQ-Principally Injective Modules, International Journal of Mathematics Archive – 3(3), 2012 , Page 962 - 967.

13. S. Wongwai and M. Kaewneam, NPQ-Injective Modules, International Journal of Mathematics Archive – 7(8), 2016, 148 - 154.

Source of support: Nil, Conflict of interest: None Declared.