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Some results on nil-injective rings
Jabbar-, Abdullah M. Abdul
*
Ferman A. Ahmed
Department of Mathematics, College of Science, Salahaddin University-Erbil, Kurdistan Region
- Iraq.
*Corresponding authors E-mail: ferman.ahmed@su.edu.krd
Received 16 Mar 2024; Received in revised form 22 Apr 2024; Accepted 27 Apr 2024, Published 30
Apr 2024
ARTICLE INFO
ABSTRACT
Keywords
Trivial extention,
nilpotent elements, nil-
injective, Wnil-
injective.
Let be a ring. A right R-module is called nil-injective if for any
element is belong to the set of nilpotent elements, and any right R-
homomorphism can be extended to . If is nil-injective, then
is called a right nil-injective ring. A right R-module is called Wnil-
injective if for each non-zero nilpotent element of , there exists a
positive integer such that that right R-homomorphism
can be extended to . If is right Wnil-injective,
then is called a right Wnil-injective ring. In the present work, we
discuss some characterizations and properties of right nil-injective and
Wnil-injective rings.
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1. Introduction
In this article, is an associative ring with identity, and All R-modules are unital. We denote
and to the right annihilator and the left annihilator of , respectively. The set of
nilpotent elements, the set of unit elements, the set of right singular elements and the Jacobson
radical of are denoted by , and , respectively. Also, by and , we
mean the set of integers modulo and integer numbers, respectively. In addition, an R-module
is called p-injective if for any principal right ideal of and any right R-homomorphism
, there exists such that , for all in , which was first introduced by Ming in
[9]. In [10] also, Yue Chi Ming generalized p-injective, which is np-injective. A right R-module
is called right np-injective if for any and any R-homomorphism can be
extended to , or equivalently, for any and any R-homomorphism
there exists such that , for all . So, the ring is called right np-injective
if is np-injective. Wei and Chen defined weakly np-injective in [7]. A right R-module is
called weakly np-injective if for any , there exists a positive integer such that
and any right R-homomorphism can be extended to . Or equivalently, for
any there exists a positive integer such that and any R-homomorphism
there exists such that , for all . If is weakly np-
injective, then is a right weakly np-injective ring. It is easy to check that every right np-injective
module is right weakly np-injective. Wei and Chen [6] generalized p-injective to nil-injective.
They have defined that a right R-module is called nil-injective, if for any and any R-
homomorphism can be extended to , or equivalently, for any
and any R-homomorphism there exists such that , for all .
So, the ring is called right nil-injective if is nil-injective. A right R-module is called Wnil-
injective if for any , there exists a positive integer such that and any
right R-homomorphism can be extended to . Or equivalently, for any
, there exists a positive integer such that and any R-homomorphism
there exists such that for all [6]. A ring is called semiprimitive
ring if [1]. We found that if is right continuous ring and is nil injective ring,
then is semiprimitive. In the matrix ring, If is a right Wnil-injective ring, for some
, then is a right nil-injective ring.
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2. Nil-Injective Rings
In this section, we consider some examples and primary results about nil-injective. Wei and Chen
[6] are poved that a ring is a right nil-injective if and only if , for every
. We found some non-tivial examples of nil-injective rings via those theorem. Recall that if
the ring of scalars is commutative, then for all and , we have . Let be a
ring and a bimodule over R. The trivial extension of and is
with addition defined componentwise and multiplication defined by
[5]. So, we obtain that for any , for , there exist such that
, then , for every and for every .
Thus, the set of nilpotent elements in is given by:
. In addition, we found some examples which are not p-injective rings but they are nil-injective
rings:
Example 2.1 Let be a ring with addition
defined componentwise and multiplication defined by . Now,
. Firstly, and
. Secondly, . So,
Thirdly, .
So, Thus, is right nil-injective ring. But, is not right
p-injective ring because Then, . So,
, but . Thus,
. Hence, is not right p-injective ring.
Example 2.2 Let be an external direct sum of and
with standard addition and multiplication. Since . Firstly,
. Secondly,
. Then, . Thus, is
right nil-injective ring. But, is not right p-injective ring because Then,
. So, , but . Thus,
. Hence, is not right p-injective ring.
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Proposition 2.3 If . Then, is right nil-injective ring.
Proof. Let . Then, . So,
.
We have two cases for find . Firstly, if is non-zero divisor, then is unit. There is
nothig to prove. Secondly, if is zero divisor,
. So,
. Therefore, , for
all Thus, is right nil-injective ring.
Proposition 2.4 Let and has non-zero nilpotent element. Then, is right nil-
injective if , for each .
Proof. Suppose that
is a ring with addition defined and
multiplication defined by
. It is clear that . We
obtain that, .
Since , then
. So,
. Therefore, for each non-
zero nilpotent element . Hence, is right nil-injective ring.
Proposition 2.5 Let be a local right nil-injective ring. Then for any non-zero (two-sided) ideals
and of R, , for any .
Proof. Suppose that and define the map by for
. Let for . So , yielding
. Thus, is well-defined. Since is right nil-injective, then can be extended on . Therefore,
, for some . Thus, . Since local, then by
[Proposition 7.2.11.,[6]] either or is a unit, but .
Thus, or , a contradiction. Hence, for any .
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Proposition 2.6 Let be a right nil-injective ring. Let :
(1) If and an idempotent generates R. Then there exists an idempotent such
that , and is a direct summand of .
(2) If and are generated by two idempotent elements with , then there exists
an idempotent such that .
Proof. (1) Suppose that , for some and , we define is
an isomorphism, then for some and , for some Now,
. Since , for some . So,
. Thus, is an idempotent. So,
. Let then , so
. But, as is a right nil-injective. Then, is an idempotent and . Now, let
then , so . But, as is
a right nil-injective, then . Therefore, and . This gives
for some . Since , we get and so , where and
. Now, is a direct summand of . We have to prove that
. Let . Then, and
. Then, . Thus, . Since is an idempotent, then . Therefore,
. Then, . Thus, . Hence, .
(2) Suppose that and for some idempotents and of . Then,
[as and
]. Now, implies . So, by (1) , ,
and .
Therefore, (since
and . Therefore,
. Thus, , where
. Hence, is a direct summand of
.
Theorem 2.7. Let be a right Wnil- injective ring. If embeds in , where , then
there exists a positive integer number such that is an image of .
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Proof. If is monic. Since is a right Wnil-injective, there exists a positive integer
such that any right R-homomorphism of into extends to one of into . Let right R-
homomorphism , where and are embedation maps.
Hence , where . Now let , via:
. Since , there exists a positive integer such that
. Since
. Let , where . Hence
and so is an epic.
Proposition 2.8. If is a nil-injective ring, then is a direct summand of , for all .
Proof. Let be a nil-injective ring and consider the row exact diagram of R-modules,
Let is the identity mapping on and is the canonical injection. If completes the
diagram commutatively, then . Hence, is a splitting map for . If , then ,
so . If , since . Then,
Thus, and . Therefore, . If
, then for some , so . Hence, and we have
. Since , then . Hence, is direct sumand of .
Definition 2.9. A given is a ring if it satisfies the following two conditions:
(1) For any right ideal there is an idempotent such that is an essential extension of .
(2) If , is isomorphic to a right ideal , then also is generated by an idempotent.
A ring is right continuous [12] if it satisfies Conditions 1 and 2.
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Lemma 2.10. [Lemma4.1, [12]] If is a right continuous ring, then , and
is regular.
Lemma 2.11. [Lemma 2.1,[11]] If Z ( ) contains no non-zero nilpotent element, then Z()=0.
The following results are about the relation between right nil-injective ring and right continuous
rings:
Proposition 2.12. Let be a ring such that is right continuous ring and is nil injective
ring. Then is semiprimitive.
Proof. By Lemma 2.10, . We shall show that If not, by Lemma
2.11, there exists then . Since a right continuous ring, then by Lemma
2.10, is nil-injective, any R-homomorphism of into extends to one of into
. Let such that where , we have to show that is
well defined, let where then . Thus,
so is well defined right
R-homomorphism, since is nil-injective, there exists such that
, then . So . Since ,
then is invertible. We get that , which is a contradiction. Therefore, . So,
. This shows that is semiprimitive.
We construct a relation between right Wnil-injective and right nil-injective in the matrix ring as
follow:
Lemma 2.13. A given ring is right Wnil-injective if and only if for any
, there exists a positive integer n such that and .
Theorem 2.14. Let be a ring and be the matrix ring. Let
, for , then the followings are true:
(1) if and only if .
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(2) If is a right Wnil-injective ring, for some , then is a right nil-injective ring.
Proof. 1. Let then . Now, take , then
,
so we have , for all . That is, so , for
, yielding . Thus, , hence .
Therefore, . So, we can write , where ,
which implies . Hence, . Conversely, Let
then . Now, if , then which implies
thus that is hence for . So,
Then, If then . So,
for . Thus, implies then
. So, with for . Thus
. Therefore,
.
(2) Let ) and take, . Now, is right Wnil-injective. So, by Lemma
2.13. there exists such that and . Since ,
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. So it must be that and
. Thus is right nil-injective.
A non-zero right R-module is said to be s-unital [4], if for each . If is s-unital,
then is called a right s-unital ring. If is a right R-module and is a subset of , then we set
. HIRANO and TOMINAGA introduced in [13], if is a right R-module and
is a subset of , then . So, if is a right R-module and is an element
of , then . Finally, if is a right R-module and is an element of ,
then .
Theorem 2.15. If is a finite subset of a right s-unital ring , then there
exists an element such that, for all .
An R-module is called right nil-injective module if each and each homomorphism
, there exists a homomorphism such that , for every [2].
Theorem 2.16 Let be s-unital module, then the following conditions are equivalent:
(1) is a right nil-injective module.
(2) for every .
(3) where , then .
(4) If , is R-linear, then .
Proof.
Assume that is nil-injective. Given such that there
exist an element such that . Then, by Theorem 2.15., there exists an element
such that and . Consider defined by . Since is a nil-
injective, we can find an element with for all . We therefore obtain
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, which means . On the other hand, let , for some
. Then, , for every . Thus, . So that .
Let . Then
Suppose such that . Then, . Therefore,
.
First, as implies that
where . Now, we must show . Then, . Therefore,
. We have,
. Let , then . Thus, . Therefore
. Now, let , then . This means that
. So, whenever showing that
and so . This implies that for some yielding
that is . Thus, . Hence,
.
Let be R-linear map with . Then, , for some .
This proves (1). Which completes the proof.
3. Conclusions
In conclusion, our study has demonstrated examples of rings that are nil-injective but not p-
injective. We also attempted to find examples of rings that are Wnil-injective but not nil-injective.
These examples highlight the importance of studying these generalizations, as they different from
the previous types of nil-injective rings.
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