Keywords:
In this paper we introduce and investigate the notion of s-injective Modules and Rings. A right R-module M is called right s-N-injective, where N is a right R-module, if every R-homomorphism
extends to
, where K is a submodule of the singular submodule
. M is called strongly s-injective if M is s-N-injective for every right R-module N. The connection between this new injectivity condition and other injectivity conditions has been established, and examples are provided to distinguish s-injectivity from other injectivity concepts such as mininjectivity, soc-injectivity. Several properties of this new class of injectivity are highlighted.
Throughout this paper all rings are associative with identity, and all modules are unitary R-modules. For a right R-module
, we denoted
,
, and
by the Jacobson radical, the socle and the singular submodule of
, respectively.
,
,
,
and
are used to indicate the right socle, the left socle, the right singular ideal, the left singular ideal, and the Jacobson radical of
, respectively. For a submodule
of
, the notations
,
and
mean, respectively, that N is essential, maximal, and direct summand. If
is a subset of a right R-module
, right annihilators will be denoted by
, with a similar definition of left annihilators
. Multiplication maps
and
will be denoted by
and
, respectively. If M and N are right R-modules, then M is called N-injective if every R-homomorphism from a submodule of N into M can be extended to an R-homomorphism from N into M. Mod-R indicates the category of right R-modules. We refer to [1-3] for all the undefined notions in this paper.
2. Strongly S-Injective Modules
Definition 1 A right R-module M is called s-N-injective if every R-homomorphism
extends to
, where
is a submodule of the singular submodule
. M is called s-injective if
is s-R-injective.
is called strongly s-injective, if
is s-N-injective for all right R-modules
.
For example every nonsingular R-module is strongly s-injective. In particular, the ring of integers
is strongly s-injective, but not injective.
Proposition 1
1) Let
be a right R-module and
a family of right R-modules. Then the direct product ![]()
is s-N-injective if and only if each
is s-N-injective, ![]()
2) Let M,
, and
be right R-modules with
If M is s-N-injective, then M is s-K-injective.
3) Let M,
, and
be right R-modules with
If
is s-M-injective, then
is s-M-injective.
4) Let
be a right R-module and
a family of right R-modules. Then
is
injective
if and only if
is
injective, ![]()
5) A right R-module
is s-injective if and only if
is s-P-injective for every projective right R-module
.
6) Let M,
, and
be right R-modules with
If
is s-K-injective, then
is s-K- injective.
7) If
,
, and
are right R-modules,
and
is s-A-injective, then
is s-B-injective.
Proof. The proofs of 1) through 4) are routine.
5). This follows from 4).
6). Let
be an R-homomorphism where
is singular submodule of
. Then the map
can be extended to an R-homomorphism
, where
the inclusion map. Now, the map
is an extension of
, where
the natural projection map into
.
7). Let
be an R-isomorphism, and
an R-homomorphism where
is a singular submodule of
. The restriction of
to
induces an isomorphism
By hypothesis, the map
can be extended to an R-homomorphism
Now, the map
is an extension of
. □
The next two corollaries are immediate consequences of the above proposition.
Corollary 1 Let
be a right R-module. Then the following statements are true:
1) A finite direct sum of s-N-injective modules is again s-N-injective. In particular a finite direct sum of s-injective (strongly s-injective) modules is again s-injective (strongly s-injective).
2) A summand of s-N-injective (s-injective, strongly s-injective) module is again s-N-injective (s-injective, strongly s-injective) module.
Corollary 2
1) Let
be a right R-module and
in
, where the
are orthogonal idempotents. Then
is s-injective if and only if
is
injective for each
, ![]()
2) Assume that
and
are idempotents of
,
and
is
injective. Then
is s-fN-injective.
Proposition 2 If
is a finitely generated right R-module, then the following conditions are equivalent:
1) Any direct sum of s-N-injective modules is s-N-injective.
2) Any direct sum of injective modules is s-N-injective.
3)
is noetherian.
Proof. 1)
2). Clear.
2)
3). Consider a chain
of singular submodules of
and
. Let
be the injective hull of
,
and
be a map defined by
. Since,
is s-N-injective,
can be extended to an R-homomorphism
Since
is finitely generated,
for some
then
and
for
every
. Hence
is noetherian.
3)
1). Let
be a direct sum of s-N-injective modules, and
be a homomorphism of
right R-modules where
. Since
is noetherian,
for some finite subset ![]()
Since finite direct sums of s-N-injective modules is s-N-injective,
can be extended to an R-homomorphism
□
The second singular submodule of a right R-module
, denoted by
, is defined by the equality
. We see that a
is closed submodule of
and
is non-singular. A right R-module is Goldie torsion if
.
Lemma 1 Let
and
be right R-modules such that
is injective. Then every homomorphism
where
extends to
.
Proof. Let
where
is injective and
. If
is a homomor- phism where
such that
, then
is singular. So
extendable
to
. Now suppose that
, so
is singular which is a contradiction. Thus
. Suppose that
where
and
,
. So
and the kernal of the map
by
is essential in
which is a contradiction. Then every homomorphism
where
extends to
. □
Proposition 3 The following statements are equivalent:
1)
is strongly s-injective.
2)
is
injective, where
is the injective hull of
.
3)
, where
is nonsingular and
is injective with
.
4)
is injective.
5)
is G-injective for every Goldie torsion module
.
6)
is I-injective, where
is the injective hull of
.
Proof. 1)
2). Clear.
2)
3). If
, we are done. Assume that
and consider the following diagram:
![]()
where
and
are inclusion maps and
is injective closure of
in
. Since
is
injective,
is s-D-injective. So there exists an R-homomorphism
which extends
Since
,
is an embedding of
in
. If we write
then
for some submodule
of
because
is injective. Finally
is nonsingular because
.
3)
4). Since
is singular and
is singular submodule of
, so
and
for some submodule
of
. Then
and
is injective.
4)
5). Let
be a Goldie torsion right R-module and
is a submodule of
. Using the above Lemma, every homomorphism
extends to
.
5)
6). If
is the injective hull of
, then
and
is Goldie torsion.
6)
1). Let
be a right R-module and
be a singular submodule of
. Consider the diagram
![]()
where
,
, and
are the inclusion maps. Since
is I-injective and
is injective. So, there exist R- homomorphisms
and
such that
and
. Thus
. Hence
is strongly s-injective. □
Corollary 3 Let
a Goldie torsion right R-module. Then
is injective if and only if
is strongly s-injective.
Proposition 4 For a ring
, the following conditions are equivalent:
1)
is strongly s-injective.
2)
is s-I(RR)-injective, where
is the injective hull of
.
3)
where
is injective and
is nonsingular. Moreover, if
, then
, and in this case
and
are relatively injective.
4)
is injective.
5)
is G-injective for every Goldie torsion right R-module
.
6)
is G-injective, where
is the injective hull of
.
7) Every finitely generated projective right R-module is strongly s-injective.
Proof. The equivalence between 1), 2), 3), 4), 5) and 6) is from Proposition 3.
1)
7) Since a finite direct sum of s-N-injective is s-N-injective for every right R-module
(Corollary 1), so every finitely generated free right R-module is strongly s-injective. But a direct summand of strongly s-injective is strongly s-injective (Corollary 1). Therefore every finitely generated projective module is strongly s-injective. The converse is clear. □
The following examples show that the two classes of strongly s-injective rings and of soc-injective rings are different.
Example 1 Let
be the field of two elements,
for
,
,
If ![]()
is the subring of
generated by
and
then
is a von Neumann regular ring with
, and hence
and
is strongly s-injective. However, the map
given by
cannot be extended to an R-homomorphism from
into
(suppose that
for some
. Then for every
,
which is impossible), and so
is not a soc-injective ring.
Example 2 Let
where
for all
for all
and
for all
and
Then
is a commutative, semiprimary, local ring with
and
has simple essential socle
. It is not difficult to see that
is right soc-injective. However the R-homomorphism
defined by
for all
can not be extended to an endomorphism of
, and so
is not s-injective ring.
Definition 2 A ring
is called a right generalized V-ring (right GV-ring) if every simple singular right R-module is injective.
Proposition 5 A ring
is right GV-ring if and only if every simple right R-module is strongly s-injective.
Proof. Let
be a right GV-ring and
be a simple right R-module. The module
is either projective or singular, so
is strongly s-injective. Conversely, if
is a simple singular right R-module, then
is strongly s-injective. Thus
is injective by Proposition 3. □
Lemma 2 For a right R-module
the following conditions are equivalent:
1)
satisfies
on essential submodules.
2)
is noetherian.
Proof. Assume that
has
on essential submodules. Then
is noetherian for every submodule
where
is an intersection complement of
and
is noetherian). In particular, every uniform submodule of
is noetherian. Let
be an intersection complement of
(see Kasch [2] p.112). Then
is noetherian. So, to prove that
is noetherian it is enough to show that
is noetherian. Assume that
contains an infinite direct sum
of nonzero submodules
. Since
, each
contains a proper essential submodule
and
is essential in
. But this gives that
is noetherian which is impossible because
with each
nonzero. Then
contains
independent uniform submodules
such that
is essential in
. Thus
and
are noetherian. Hence
is noetherian. □
It is well-known that, a ring
is right noetherian if and only if all direct sums of injective right R-modules are injective. In the next Proposition we obtain a characterization of ring which has
on essential right ideals in terms of strongly s-injective right R-modules.
Proposition 6 The following conditions on a ring
are equivalent:
1) Every direct sum of strongly s-injective right R-modules is strongly s-injective.
2) Every direct sum of injective right R-modules is strongly s-injective.
3) Every finitely generated right R-module has
on essential submodules.
4)
is noetherian.
Proof. 1)
2). Clear.
2)
3). Consider a chain
of essential submodules of a finitely generated right R-module
and
. Let
be the injective hull of
and
be a map defined by
. Since
is strongly s-injective and
has an essential singular submodule, so
is injective and
can be extended to an R-homomorphism
Then
for some
and
. Thus ![]()
for every
. Hence
has
on essential submodules.
3)
4). Above Lemma.
4)
1). Let
be a direct sum of injective Goldie torsion modules. Then
,
, may be
considered as an injective R/Sr-module. Thus
is an injective R/Sr-module. Now, suppose that ![]()
is a nonzero map where
is simple right ideal, so
is a singular right ideal and
where
is finite. Thus
extends to
. Then
which is a contradiction
with
. Hence any homomorphism
where
is a right ideal of
induces a map
with
. The map
extends to a homomorphism
. Then the map
where π is the natural epimorphism
, extends
. Hence
is injective. Therefore, every direct sum of strongly s-injective right R-modules is strongly s-injective. □
If
is an ideal of
,
is called I-semiperfect if for every right ideal
, there is a decomposition
such that
and
[4].
Lemma 3 If
is Zr-semiperfect, then the following statements hold:
1) A module
is s-injective if and only if
is injective.
2)
for all singular right ideals
of
if and only if
for all right ideals
of
.
Proof. 1) Let
be s-injective, and
be an R-homomorphism where
is a right ideal of
. Then
where
Let
be an extension of the restriction map
Define
by
for all
Clearly,
is an extension of
, and so
is injective by the Baer's Criterion.
2) Let
be a right ideal of
. Since
is right
-semiperfect, then
where ![]()
and
. So
and
. If
, then
and so
. The last equality is because that
is a singular right ideal of
. Write
where
. Then
. Therefore,
. □
Proposition 7 Let
be a right R-module.
is semisimple summand of
if and only if every right R-module is s-M-injective.
Proof. If
is semisimple summand of
, then every right R-module
is s-M-injective. Conversely, if every right R-module is s-M-injective, then every identity map
where
is singular submodule of
extends to
. Thus
is a summand of
. Hence
is a semisimple summand of
. □
Corollary 4 A ring
is right nonsingular if and only if every right R-module is s-injective.
A ring
is called a right (left)
ring if every singular right (left) R-module is injective.
rings were initially introduced and investigated by Goodearl [5]
Theorem 1 The following statements are equivalent:
1)
is right
ring.
2) Every right R-module is strongly s-injective.
3) Every singular right R-module is strongly s-injective.
Proof. Clear from Proposition 3. □
A module
is said to satisfy the C2-condition, if
and
are submodules of
,
and
then
We also say
satisfies the C3-condition if
and
are submodules of
with
and
then
is a summand of
. It is a well-know fact that the C2-condition implies the C3-condition. In the next proposition we show that s-quasi-injective modules inherit a weaker version of these conditions.
Proposition 8 Suppose
is a s-quasi-injective right R-module.
1)
If
and
are singular submodules of
,
and
then ![]()
2)
Let
and
be singular submodules of
with
If
and
then
is a summand of
.
Proof. 1) Since
and
is s-injective, being a summand of the s-quasi-injective right R-module
is s-injective. If
is the inclusion map, the identity map
has an extension
such that
and so
is a summand of
.
2) Since
and
are summands of
, and
is s-quasi-injective, both
and
are s-M- injective. Thus the singular module
is s-M-injective, and so is a summand of
. □
It is a well-known fact that the C2-condition implies the C3-condition.
Proposition 9 If a module
has s-C2-condition, then
has s-C3-condition.
Proof. Consider singular summands
and
of
such that
. Write
and let π denote the projection
. Then
. If
,
where
and
and
,
and
. Then
is a mono-
morphism; so
is a summand of
by
-
. As
,
.
Proposition 10 Let
and
be Morita-equivalent rings with category equivalence
Let
,
, and
be right R-modules. Then
1)
is singular if and only if
is singular.
2)
is s-N-injective if and only if
is s-injective.
Proof. There is a natural isomorphisms
and
. This means that for each
there is an isomorphism
in
such that for each
,
in
and each
in
, the following diagram commutes
![]()
1). The right R-module
is singular if and only if there is an exact sequence of right R-modules
with essential monomorphism
. But using [6, Proposition 21.4 and Proposition 21.6] the sequence
is exact with essential monomorphism
if and only if the sequence
of right S-modules is exact with essential monomorphism
. So
is singular if and only if
is singular.
2). Let
be a s-N-injective and
be a singular submodule of
. Let
be a homomorphism. Since
is singular,
is a monomorphism and the maps
and
are isomorphisms ( we may assume that
is a submodule of
), then we have the commutative diagram
![]()
So the following diagram
![]()
is commutative where
is
injective. The converse is similarly. □
As for right self-injectivity, right strongly s-injectivity turns out to be a Morita invariant.
Theorem 2 Right strong s-injectivity is a Morita invariant property of rings.
Proof. Let
and
be Morita-equivalent rings with category equivalences
and
. Let
, and
be right R-modules.
is finitely generated projective R-module if and only if
is finitely generated projective S-module [6, Propositions
and
]. Also
is s-N-injective if and only if
is s-F(N)-injective (Proposition 10) and then
is strongly s-injective if and only if
is strongly s-injective. Then, every finitely generated projective right R-module is strongly s-injective if and only if every finitely generated projective right S-module is strongly s-injective. Therefore right strong s-injectivity is a Morita invariant property of rings. □
Proposition 11 For a projective right R-module
, the following conditions are equivalent:
1) Every homomorphic image of a s-M-injective right R-module is s-M-injective.
2) Every homomorphic image of an injective right R-module is s-M-injective.
3) Every singular submodule of
is projective.
Proof. 1)
2) Obvious.
2)
3) Consider the following diagram:
![]()
Where
is a singular submodule of
,
and
are right R-modules with
injective,
an R-epimorphism, and
an R-homomorphism. Since
is s-injective,
can be extended to an R-homomo- rphism
Since
is projective,
can be lifted to an R-homomorphism
such that
Now, define
by
Clearly,
and hence
is projective.
3)
1) Let
and
be right R-modules with
an R-epimorphism,
is a singular submodule of
and
is s-M-injective. Consider the following diagram:
![]()
Since
is projective,
can be lifted to an R-homomorphism
such that
Since
is s-injective,
can be extended to an R-homomorphism
Clearly,
extends
. □
Corollary 5 The following conditions are equivalent:
1) Every quotient of s-injective right R-module is s-injective.
2) Every quotient of an injective right R-module is s-injective.
3) Every singular right ideal is projective.
Proposition 12 The following conditions are equivalent:
1) Every strongly s-injective right R-module is injective.
2) Every nonsingular right R-module is semisimple injective.
3) For every right R-module
,
where
semisimple injective.
4)
is
semiperfect.
Proof. 1)
2) If
is nonsingular right R-module, then
is strongly s-injective. Thus
is semisimple injective.
2)
3) Let
be a right R-module. If
is Goldie torsion, we are done. Now suppose that
is not essential in
and
be a maximal submodule
with respect to
. Then
is semisimple injective and
. It is clear that
.
3)
4). Let
be a right ideal of
. Then
where
is semisimple injective. We have
and
is a summand of
and generated by an idempotent. Hence
is
semiperfect.
4)
1). Let
be a strongly s-injective and
be an R-homomorphism where
is a right ideal of
By
where
and
Using Proposition 3 let
be an extension of the restriction map
Define
by
for all
Clearly,
is an extension of
, and so
is injective by the Baer's Criterion. □
Theorem 3 The following are equivalent for a ring
:
1)
is a right PF-ring.
2)
is Zr-semiperfect, right strongly s-injective ring with essential right socle.
3)
is semiperfect, right min-C2, right strongly s-injective ring with essential right socle.
4)
is semiperfect with
, right strongly s-injective ring with essential right socle.
5)
is right finitely cogenerated, right min-C2, right strongly s-injective ring.
6)
is a right Kasch, right strongly s-injective ring.
7)
is a right strongly s-injective ring and the dual of every simple left R-module is simple.
Proof. 1)
2). Clear.
2)
1). Clear by Lemma 2.
1)
3). Clear.
3)
4). Suppose that
is a non singular simple right ideal in
, so
for some simple right ideal
with
. Thus
and
is a summand which is a contradiction. Then
. The other inclusion is clear.
4)
1). Let
be semiperfect and right strongly s-injective ring with essential right socle and
. Then
where
is injective and
. Thus
and
. The right ideal
may be considered as an R/J-module. Let
be a map where
is a right ideal of
,
induces a map
given by
. Since
is injective as an R/J-module so
extends to
. The map
, where π is the natural epimorphism
, extends
and
is injective. Therefore
is right selfinjective and
is right
.
1)
5). Clear.
5)
1). Since
is right strongly s-injective ring, it follows from Proposition 3 that
, where
is injective and
is nonsingular. If
is simple right ideal in
such that
, then, by the proof of 3)
1)
and every simple right ideal in
is a summand of
. Since
is right finitely cogenerated,
has a finitely generated essential socle. Thus
is semisimple. Hence
is injective and
is right
ring.
1)
6) Proposition 3 and [7, Theorem 5]
1)
7). Assume that
is right strongly s-injective and the dual of every simple left R-module is simple. If
is a nonsingular simple right ideal, then
for some simple right ideal
with
. But
is
, so
and
is a summand. Thus
is right min-CS, so by [3, Theorem 4.8]
is semiperfect with essential right socle. Hence
is right
by 3). □
The following is an example of a right perfect, left Kasch ring and right strongly s-injective which is not right self-injective ring.
Example 3 Let
be a field and let
be the ring of all upper triangular, countably infinite square matrices over
with only finitely many off-diagonal entries. Let
be the K-subalgebra of
generated by
and
Then
is a right perfect, left Kasch (
has only one simple left R-module
up to isomorphism. So
) such that
whereas
is neither left perfect nor right self-injective because it is not right finite dimensional.
Remark 1 Note that the ring of integers
is an example of a commutative noetherian strongly s-injective ring which is not quasi-Frobenius.
Definition 3 A ring
is called right CF-ring (FGF-ring) if every cyclic (finitely generated) right R-module embeds in a free module. It is not known whether right CF-rings (FGF-rings) are right artinian (quasi- Frobenius). In the next result we provide a positive answer if we assume in addition that the ring
is strongly right s-injective.
Proposition 13 Every right
right strongly s-injective ring is quasi-Frobenius.
Proof. Theorem 3 and [7, Theorem 5] □
3. S-CS Modules and Rings
A module
is said to satisfy C1-condition or called
module if every submodule of
is essential in a direct summand of
.
Definition 4 A right R-module
is called s-CS module if every singular submodule of
is essential in a summand of
.
For example, every nonsingular module is s-CS. In particular, the ring of integers
is s-CS but not ![]()
Proposition 14 For a right R-module
, the following statements are equivalent:
1) The second singular submodule
is
and a summand of
.
2)
is s-CS.
Proof. 1)
2). If the second singular submodule
of
is
and a summand of
, then every singular submodule of
is a summand of
and a summand of
.
2)
1). Let
be s-CS and
is a submodule of
. Then
where
is a sum-
mand of
and
. But
is closed, so
. Since
and
is closed in
, so
and
is
. In particular,
is the only closure of
. Thus
is a summand of
. □
A module is called s-continuous if it satisfies both the s-C1- and s-C2-conditions, and a module is called quasi-s-continuous if it satisfies the s-C1- and s-C3-conditions, and
is called a right s-continuous ring (right quasi-s-continuous ring) if
has the corresponding property. Clearly every strongly s-injective is s-continuous.
Proposition 15 If every singular simple right R-module embeds in
and
is s-CS, then
is finitely cogenerated.
Proof. Let
be a s-CS and every singular simple right R-module embeds in
. Then
is a
and summand of
by above Proposition. Also
cogenerats every simple quotient of
then by [3, Theorem 7.29],
is finitely cogenerated.
Proposition 16 Let
be a ring. Then
is a right PF-ring if and only if
is a cogenerator and
is
.
Proof. Every right PF-ring is right self-injective and is a right cogenerator by [3, 1.56]. Conversely, if
is a
and
is cogenerator then
has finitely generated, essential right socle by Proposition 15. Since
is right finite dimensional and
is a cogenerator, let
and
be the injective hull of
, then there exists an embedding
for some set
. Then
for some projection
, so
and hence is monic. Thus
is monic, and so
where
is nonsingular. So
is a right PF-ring by Theorem 3. □
Proposition 17 If every simple singular right R-module embeds in
and
is continuous, then
is semiperfect.
Proof. Let
be continuous and every simple singular right R-module embeds in
. Then
has a finitely generated essential socle by Proposition 15. Thus, by hypothesis, there exist simple submodules
of
such that
is a complete set of representatives of the isomorphism classes of
simple singular right R-modules. Since
is
, there exist submodules
of
such that
is a direct summand of
and
for
. Since
is an indecomposable continuous R-module, it has a local endomorphism ring; and since
is projective,
is maximal and small in
by [3, 1.54]. Then
is a projective cover of the simple module
. Note that
clearly implies
; and the converse also holds because every module has at most one projective cover up to isomorphism. But it is clear that
if and only if
if and only if
. Moreover, every
is singular. Thus,
is a complete set of representatives of the isomorphism classes of simple singular right R-modules. Hence every simple singular right R-module has a projective cover. Since every non-singular simple right R-module is projective, we conclude that
is semiperfect. □
References