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International Journal of Pure and Applied Mathematics

Volume 106 No. 2 2016, 543-550

ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.eudoi: 10.12732/ijpam.v106i2.17

PAijpam.eu

ON 2-ABSORBING δ-PRIMARY

GAMMA-IDEAL OF GAMMA RING

Mohamed Youssfi Elkettani1, Abdulbakee Kasem2 §

1,2Department of Mathematics Faculty of SciencesUniversity Ibn Tofail

B.P 242, Kenitra, MOROCCO

Abstract: In this paper, the notion of 2-absorbing δ-primary Γ-ideal of Γ-ring is introduced

which unify 2-absorbing Γ-ideal and 2-absorbing δ-primary Γ-ideal , and several properties

are investigated. Here δ is a mapping that assigns to each Γ-ideal J a Γ-ideal δ(J) of the same

Γ-ring such that:(1) (∀I ∈ J(M))(I ⊆ δ(I)),(2) (∀I, J ∈ J(M)))(I ⊆ J ⇒ δ(I) ⊆ δ(J)), where

J(M) is the set of Γ-ideal of Γ-ring M .

AMS Subject Classification: 13A15

Key Words: Γ-ideal, intersection preserving, global Γ-ideal expansion, Γ-ring homomor-

phism, 2-absorbing δ-primary

1. Introduction

The notion of a Γ-ring is introduced by Nobusawa in[6], as more general thana ring. In [1] Barnes weakened slightly the conditions in the definition of theΓ-ring in the sense of Nobusawa. Barnes[1], Kyuno [4] and Luh[5] studied thestructure of Γ-rings and obtained various generalizations analogous to corre-sponding parts in ring theory. Prime ideals and primary ideals are two of themost important structures in ring theory. In [8] Zhao investigated the possibil-

Received: November 5, 2015

Published: February 15, 2016

c© 2016 Academic Publications, Ltd.

url: www.acadpubl.eu

§Correspondence author

544 M.Y. Elkettani, A. Kasem

ity of a unified approach to studying such two ideals, and introduced the notionof δ-primary ideals for a mapping δ that assigns to each ideal I an ideal δ(I) ofthe same ring. Such δ-primary ideals unify the prime and primary ideals underone frame. In [2] and [3], The concepts of prime ideal and primary ideal wasextended to the context of 2-absorbing ideal and 2-absorbing primary ideal. In[7] Jun and all are trying to apply the Zhao’s idea in ring theory to a Γ-ring. Inthis paper, we introduced the notion of 2-absorbing δ-primary Γ-ideal of Γ-ringwhich unify 2-absorbing Γ-ideal and 2-absorbing δ-primary Γ-ideal. A numberof results concerning 2-absorbing δ-primary Γ-ideals of Γ-ring are given.

2. Preliminaries

Recall that if M and Γ be two Abelian groups and for all x, y ∈ M and allα, β ∈ Γ the conditions: 1. xαy ∈ M ;2. (x+ y)αz = xαz + yαz, x(α+ β)z = xαz + xβz, xα(y + z) = xαy + xαz;3. (xαy)βz = xα(yβz);are satisfied, then we call M a Γ-ring. By a right (resp. left) Γ-ideal of a Γ-ring M we mean an additive subgroup U of M such that U ΓM ⊆ U (resp.MΓU ⊆ U). If U is both a right and a left Γ-ideal, then we say that U is aΓ-ideal of M. A Γ-ideal I of M is said to be prime if for any ideals U and V ofM , UΓV ⊆ I implies U ⊆ I or V ⊆ I. We note from [1] that a proper Γ-idealI of M is prime if a aΓb ⊆ I implies a ∈ I or b ∈ I for all a, b ∈ M . A mappingσ : M → M

′

of Γ-rings is called a Γ-ring homomorphism if it satisfies:1. σ(a+ b) = σ(a) + σ(b) for all a, b ∈ M ;2.σ(aγb) = σ(a)γσ(b) for all a, b ∈ M and γ ∈ ΓFor more detail see for example [1] and [7].

Definition 2.1. [7] A Γ-ideal I of M is said to be primary if it satisfies:

(∀a, b ∈ M)(∀γ ∈ Γ)(aγb ∈ I, a /∈ I ⇒ b ∈√I)

Denote by J(M) the set of all Γ-ideal of M

Definition 2.2. [7] An expansion of Γ-ideal in M is defined to be afunction δ : J(M) → J(M) such that:

1. (∀I ∈ J(M))(I ⊆ δ(I));

2. (∀I, J ∈ J(M)))(I ⊆ J ⇒ δ(I) ⊆ δ(J)).

Example 1. (1) The identity function Id : J(M) → J(M) is a Γ-idealexpansion of M . (2) The constant function c : J(M) → J(M), I 7→ M , is a

ON 2-ABSORBING δ-PRIMARY... 545

Γ-ideal expansion of M . (3)The radical of Γ-ideal,√

: J(M) → J(M), I 7→√I

is a Γ-ideal expansion of M

Definition 2.3. [7] Given a Γ-ideal expansion δ of M , a Γ-ideal I ∈ J(M)is said to be δ-primary if it satisfies:

(∀a, b ∈ M)(∀γ ∈ Γ)(aγb ∈ I, a /∈ I ⇒ b ∈ δ(I))

Note that the following notions are defined in commutative ring M .

Definition 2.4. [2] A proper ideal I of M is called a 2-absorbing ideal ofM if whenever a, b, c ∈ M and abc ∈ I, then ab ∈ I or ac ∈ I or bc ∈ I.

Definition 2.5. [3] A proper ideal I of M is said to be a 2-absorbingprimary ideal of M if whenever a, b, c ∈ R with abc ∈ I, then ab ∈ I or ac ∈

√I

or bc ∈√I

we can give the following definitions, they are analogue of definitions 2.4and 2.5 in the Context of Γ-ring.

Definition 2.6. A proper Γ-ideal I of Γ-ring M is called a 2-absorbingΓ-ideal of M if whenever a, b, c ∈ M , α, β ∈ Γ and aαbβc ∈ I, then aαb ∈ I oraβc ∈ I or bβc ∈ I.

Definition 2.7. A proper Γ-ideal I of Γ-ring M is called a 2-absorbingprimary Γ-ideal of M if whenever a, b, c ∈ M , α, β ∈ Γ and aαbβc ∈ I, thenaαb ∈ I or aβc ∈

√I or bβc ∈

√I.

Remark 1. every 2-absorbing Γ-ideal of Γ-ring M is 2-absorbing primaryΓ-ideal of Γ-ring M .

3. 2-Absorbing δ-Primary Γ-Ideal

In this section, we investigate 2-absorbing δ-primary Γ-ideal of Γ-ring M whichunify 2-absorbing Γ-ideal and 2-absorbing primary Γ-ideal ofM . In what followslet M denote a Γ-ring.

Definition 3.1. Given a Γ-ideal expansion δ of M , a Γ-ideal I ∈ J(M) issaid to be 2-absorbing δ-primary if it satisfies:

(∀a, b, c ∈ M)(∀α, β ∈ Γ)(aαbβc ∈ I ⇒ aαb ∈ I or aβc ∈ δ(I) or cβb ∈ δ(I))

Example 2. (1) If δ(I) = Id(I), 2-absorbing δ-primary Γ-ideal is just2-absorbing Γ-ideal as defined in definition 2.6.


(2) If δ(I) =√I, 2-absorbing δ-primary Γ-ideal is just 2-absorbing primary

Γ-ideal as defined in definition 2.7.

In the following we will give a list of results, they are an extension of someresults in [7].

Theorem 3.2. Let δ and γ be Γ-ideal expansions of M . If δ(I) ⊆ γ(I)for all I ∈ J(M), then every 2-absorbing δ-primary Γ-ideal is also 2-absorbingγ-primary.

Proof. Let I be an 2-absorbing δ-primary Γ-ideal of M . Let a, b, c ∈ M andα, β ∈ Γ be such that aαbβc ∈ I. Then aαb ∈ I or aβc ∈ δ(I) or bβc ∈ δ(I).Since I ⊆ δ(I) ⊆ γ(I) by assumption. Hence I is a 2-absorbing γ-primaryΓ-ideal of M .

Theorem 3.3. Let δ be a Γ-ideal expansion of M . For any subset S of

M , denote by Jδ(S) the intersection of all 2-absorbing δ-primary Γ-ideals of Mcontaining S. Then the function h : J(M) → J(M) given by h(I) = Jδ(I) for

all I ∈ J(M) is a Γ-ideal expansion of M .

Proof. Proof. I ⊆ Jδ(I) = h(I) for all I ∈ J(M). Let I, J ∈ J(M) be suchthat I ⊆ J . Then

h(I) = Jδ(I) =⋂

∈ {H ∈ J(M)|I ⊆ H and H is 2−absorbing δ−primary}

⊆⋂

{H ∈ J(M)|J ⊆ H and H is 2−absorbing δ−primary} = Jδ(J) = h(J).

Theorem 3.4. Let δ be a Γ-ideal expansion of M . If {Ji|i ∈ Λ} is a

directed collection of 2-absorbing δ-primary Γ-ideals of M , where Λ is an index

set, then the Γ-ideal J :=⋃

i∈Λ Ji is 2-absorbing δ-primary.

Proof. Let a, b, c ∈ M and α, β ∈ Γ be such that aαbβc ∈ J . Then thereexists Ji such that aαbβc ∈ Ji. Since Ji is 2-absorbing δ-primary and Ji ⊆ J ,it follows that aαb ∈ Ji or aβc ∈ δ(Ji) or bβc ∈ δ(Ji). Since Ji ⊆ δ(Ji) ⊆ δ(J),aαb ∈ J or aβc ∈ δ(J) or bβc ∈ δ(J), so that J is 2-absorbing δ-primary.

Recall that A Γ-ideal expansion δ is said to be intersection preserving if itsatisfies:

(∀I, J ∈ J(M))(δ(I ∩ J) = δ(I) ∩ δ(J))


A Γ-ideal expansion δ is said to be global if for each Γ-ring homomorphismσ : M → M

′

of Γ-rings, the following holds:

(∀I ∈ J(M′

)(δ(σ−1(I)) = σ−1(δ(I)))

Note that the Γ-ideal expansion Id of M in Example 1(1) is both intersectionpreserving and global.

Example 3. [7, Theorem 8] For each I ∈ J(M), let

B :=⋂

{J |I ⊆ J and J is a prime Γ− ideal of M}

Then a function δ : J(M) → J(M) given by δ(I) = B(I) for all I ∈ J(M) is anintersection preserving Γ-ideal expansion of M .

Theorem 3.5. Let δ be a Γ-ideal expansion of M which is intersection

preserving. If I1, I2, ..., In are 2-absorbing δ-primary Γ-ideals of M and J =δ(Ik) for all k = 1, 2, ..., n, then I :=

⋂nk=1 Ik is an 2-absorbing δ-primary Γ-

ideal of M .

Proof. Obviously, I :=⋂n

k=1 Ik is a Γ-ideal of M . Let a, b, c ∈ M andα, β ∈ Γ be such that aαbβc ∈ I and aαb /∈ I. Then aαb /∈ Ik for somek ∈ {1, 2, ..., n}. But aαbβc ∈ I ⊆ Ik and Ik is 2-absorbing δ-primary, whichimply that aβc ∈ δ(Ik) or bβc ∈ δ(Ik) Since δ is intersection preserving ,wehave

δ(I) = δ(

n⋂

k=1

Ik) =

n⋂

k=1

δ(Ik) = J = δ(Ik)

and so aβc ∈ δ(I) or bβc ∈ δ(I) Therefore I is an 2-absorbing δ-primary Γ-idealof M .

Recall that, σ : M → M′

be a Γ-ring homomorphism of Γ-rings. Note thatif J is a Γ-ideal of M

′

, then σ−1(J) is a Γ-ideal of M , and that if σ is surjectiveand I is a Γ-ideal of M , then σ(I) is a Γ-ideal of M

′

.

Theorem 3.6. Let δ be a Γ-ideal expansion which is global and let σ :M → M

′

be a Γ-ring homomorphism of Γ-rings. If J is an 2-absorbing δ-primary Γ-ideal of M

′

, then σ−1(J) is an 2-absorbing δ-primary Γ-ideal of M .

Proof. Let a, b, c ∈ M and α, β ∈ Γ be such that aαbβc ∈ σ−1(J). Thenσ(a)ασ(b)βσ(c) ∈ J , which imply that σ(a)ασ(b) ∈ J or σ(a)βσ(c) ∈ δ(J)or σ(b)βσ(c) ∈ δ(J). Since δ is global, it follows that aαb ∈ σ−1(J)) or aβb ∈σ−1(δ(J)) = δ(σ−1(J)) or bβc ∈ δ(σ−1(J)) hance σ−1(J) is 2-absorbing δ-primary.


Recall that, if σ : M → M′

is a Γ-ring homomorphism of Γ-rings, thenσ−1(σ(I)) = I for any I ∈ J(M) that contains ker(σ).

Theorem 3.7. Let σ : M → M′

be a surjective Γ-ring homomorphism of

Γ-rings and let I be a Γ-ideal of M that contains ker(σ). Then I is 2 -absorbing

δ-primary if and only if σ(I) is an 2 -absorbing δ-primary Γ-ideal of M′

, where

δ is a global Γ-ideal expansion.

Proof. If σ(I) is an 2 -absorbing δ-primary Γ-ideal of M′

, then I is 2 -absorbing δ-primary by I = σ−1(σ(I)) and Theorem 3.6. Suppose that I is 2-absorbing δ-primary. Let x, y, z ∈ M

′

and α, β ∈ Γ be such that xαyβz ∈ σ(I).Since σ is surjective, we have σ(a) = x , σ(b) = y and σ(c) = z for somea, b, c ∈ M . Then σ(aαbβc) = σ(a)ασ(b)βσ(c) = xαyβz ∈ σ(I), which implythat aαbβc ∈ σ−1(σ(I)) = I. Since I is 2 -absorbing δ-primary, it follows thataαb ∈ I or aβc ∈ δ(I) or bβc ∈ δ(I) so that xαy ∈ σ(I) or xβz ∈ σ(δ(I)) oryβz ∈ σ(δ(I)). Using the fact that δ is global , we have

δ(I) = δ(σ−1(σ(I))) = σ−1(δ(σ(I)))

and so σ(δ(I)) = σ(σ−1(δ(σ(I)))) = δ(σ(I)) since σ is surjective. Thereforeσ(I) is 2 -absorbing δ-primary.

The following result, it is an extension of [3, lemma 2.18] .

Lemma 3.8. Let δ be a Γ-ideal expansion of M and I be a 2-absorbing δ-primary Γ-ideal of M and suppose that aαbβJ ⊆ I for some elements a, b ∈ Mand some ideal J ofM and α, β ∈ Γ. If aαb /∈ I, then aβJ ⊆ δ(I) or bβJ ⊆ δ(I).

Proof. Suppose that aβJ * δ(I) and bβJ * δ(I). Then aβj1 /∈ δ(I) andbβj2 /∈ δ(I) for some j1, j2 ∈ J . Since aαbβj1 ∈ I and aαb /∈ I and aβj1 /∈ δ(I),we have bβj1 ∈ δ(I). Since aαbβj2 ∈ I and aαb /∈ I and bβj2 /∈ δ(I), we haveaβj2 ∈ δ(I). Now, since aαbβ(j1 + j2) ∈ I and aαb /∈ I, we have aβ(j1 + j2) ∈δ(I) or bβ(j1+j2) ∈ δ(I). Suppose that aβ(j1+j2) = aβj1+aβj2 ∈ δ(I). Sinceaβj2 ∈ δ(I), we have aβj1 ∈ δ(I), a contradiction. Suppose that bβ(j1 + j2) =bβj1 + bβj2 ∈ δ(I). Since bβj1 ∈ δ(I), we have bβj2 ∈ δ(I), a contradictionagain. Thus aβJ ⊆ δ(I) or bβJ ⊆ δ(I).

The following result, it is an extension of [3, theorem 2.19].

Theorem 3.9. Let δ be a Γ-ideal expansion of M and I be a proper

Γ-ideal of M . Then I is a 2-absorbing δ-primary Γ-ideal if and only if whenever

I1ΓI2ΓI3 ⊆ I for some Γ-ideals I1, I2, I3 of M , then I1ΓI2 ⊆ I or I2ΓI3 ⊆ δ(I)or I1ΓI3 ⊆ δ(I).


Proof. Suppose that whenever I1ΓI2ΓI3 ⊆ I for some Γ-ideals I1, I2, I3 ofM , then I1ΓI2 ⊆ I or I2ΓI3 ⊆ δ(I) or I1ΓI3 ⊆ δ(I). Then clearly I is a2-absorbing δ-primary Γ-ideal of M by definition. Conversely, suppose that Iis a 2-absorbing δ-primary Γ-ideal of M and I1ΓI2ΓI3 ⊆ I for some Γ-idealsI1, I2, I3 of M , such that I1ΓI2 * I. We show that I1ΓI3 ⊆ δ(I) or I2ΓI3 ⊆ δ(I).Suppose that neither I1ΓI3 ⊆ δ(I) nor I2ΓI3 ⊆ δ(I). Then there are q1 ∈ I1 andq2 ∈ I2 such that neither q1βI3 ⊆ δ(I) nor q2βI3 ⊆ δ(I). Since q1αq2βI3 ⊆ Iand neither q1βI3 ⊆ δ(I) nor q2βI3 ⊆ δ(I), we have q1αq2 ∈ I by Lemma 3.8.Since I1ΓI2 * I, we have aαb * I for some a ∈ I1, b ∈ I2. Since aαbβI3 ⊆ Iand aαb /∈ I, we have aβI3 ⊆ δ(I) or bβI3 ⊆ δ(I) by Lemma 3.8 . We considerthree cases.Case one: Suppose that aβI3 ⊆ δ(I), but bβI3 * δ(I). Since q1αbβI3 ⊆ I andneither bβI3 ⊆ δ(I) nor q1βI3 ⊆ δ(I), we conclude that q1αb ∈ I by Lemma3.8. Since (a + q1)αbβI3 ⊆ I and aβI3 ⊆ δ(I), but q1βI3 * δ(I), we concludethat (a + q1)βI3 * δ(I). Since neither bβI3 ⊆ δ(I) nor (a + q1)βI3 ⊆ δ(I), weconclude that (a+ q1)αb ∈ I by Lemma 3.8. Since (a+ q1)αb = aαb+ q1αb ∈ Iand q1αb ∈ I, we conclude that aαb ∈ I, a contradiction.Case two: Suppose that bβI3 ⊆ δ(I), but aβI3 * δ(I). Since aαq2βI3 ⊆I and neither aβI3 ⊆ δ(I) nor q2βI3 ⊆ δ(I), we conclude that aαq2 ∈ I.Since aα(b + q2)βI3 ⊆ I and bβI3 ⊆ δ(I), but q2βI3 * δ(I), we conclude that(b + q2)βI3 * δ(I). Since neither aβI3 ⊆ δ(I) nor (b + q2)βI3 ⊆ δ(I), weconclude that aα(b+ q2) ∈ I by Lemma 3.8. Since aα(b+ q2) = aαb+ aαq2 ∈ Iand aαq2 ∈ I, we conclude that aαb ∈ I , a contradiction.Case three: Suppose that aβI3 ⊆ δ(I) and bβI3 ⊆ δ(I). Since bβI3 ⊆ δ(I)and q2βI3 * δ(I), we conclude that (b+q2)βI3 * δ(I). Since q1α(b+q2)βI3 ⊆ Iand neither q1βI3 ⊆ δ(I) nor (b+q2)βI3 ⊆ δ(I), we conclude that q1β(b+q2) =q1βb + q1βq2 ∈ I by Lemma 3.8. Since q1βq2 ∈ I and q1α(b + q1)βq2 ∈ I, weconclude that q1βb ∈ I. Since aβI3 ⊆ δ(I) and q1βI3 * δ(I), we conclude that(a + q1)βI3 * δ(I). Since (a + q1)αq2βI3 ⊆ I and neither q2βI3 ⊆ δ(I) nor(a+ q1)βI3 ⊆ δ(I), we conclude that (a+ q1)αq2 = aαq2+ q1αq2 ∈ I by Lemma3.8. Since q1αq2 ∈ I and aαq2 + q1αq2 ∈ I, we conclude that aαq2 ∈ I. Now,since (a+ q1)α(b+ q2)βI3 ⊆ I and neither (a+ q1)βI3 ⊆ δ(I) nor (b+ q2)βI3 ⊆δ(I), we conclude that (a + q1)α(b + q2) = aαb + aαq2 + q1αb + q1αq2 ∈ I byLemma 3.8. Since aαq2, q1αb, q1αq2 ∈ I, we have aαq2+q1αb+q1αq2 ∈ I. Sinceaαb+ aαq2 + q1αb+ q1αq2 ∈ I and aαq2 + bαq1 + q1αq2 ∈ I, we conclude thataαb ∈ I, a contradiction. Hence I1ΓI3 ⊆ δ(I) or I2ΓI3 ⊆ δ(I).


Acknowledgements

We would like to thank the referee for his/her great effort in proofreading themanuscript.

References

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[3] A. Badawi, U. Tekir, and E. Yetkin, On 2-absorbing primary ideals in commutative rings,Bull. Korean Math. Soc. , 51, No.4 (2014), 1163-1173.

[4] S. Kyuno, On prime gamma rings, Pacific J. Math., 75, No. 1 (1978),185-190.

[5] Luh, On the theory of simple Γ-rings, Michigan Math. J., 16 (1969),65-75.

[6] N. Nobusawa, On a generalization of the ring theory, Osaka J. Math., 1 (1964), 81-89.

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