Weakly J-ideals of Commutative Rings
aa r X i v : . [ m a t h . A C ] F e b WEAKLY J-IDEALS OF COMMUTATIVE RINGS
HANI A. KHASHAN AND ECE YETKIN CELIKEL
Abstract.
Let R be a commutative ring with non-zero identity. In this paper,we introduce the concept of weakly J -ideals as a new generalization of J -ideals. We call a proper ideal I of a ring R a weakly J -ideal if whenever a, b ∈ R with 0 = ab ∈ I and a / ∈ J ( R ), then a ∈ I . Many of the basicproperties and characterizations of this concept are studied. We investigateweakly J -ideals under various contexts of constructions such as direct products,localizations, homomorphic images. Moreover, a number of examples andresults on weakly J -ideals are discussed. Finally, the third section is devotedto the characterizations of these constructions in an amagamated ring alongan ideal. Introduction
We assume throughout the whole paper, all rings are commutative with a non-zero identity. For any ring R , by U ( R ) , N ( R ) and J ( R ), we denote the set of allunits in R , the nilradical and the Jacobson radical of R, respectively. In 2017,Tekir et al., [11] introduced the concept of n -ideals. A proper ideal I of a ring R is called an n -ideal if whenever a, b ∈ R and ab ∈ I such that a / ∈ N ( R ), then b ∈ I . Recently, Khashan and Bani-Ata in [8] introduced the notion of J -ideals asa generalization of n -ideals in commutative rings, as follows: A proper ideal I of aring R is called a J -ideal if whenever a, b ∈ R with ab ∈ I and a / ∈ J ( R ), then b ∈ I .In [9], the authors generalized J -ideals and defined quasi J -ideals as those ideals I for which √ I = { x ∈ R : x n ∈ I for some n ∈ N } are J -ideals. In this paper, wedefine and study weakly J -ideals of commutative rings as a new generalization of J -ideals. We call a proper ideal I of R a weakly J -ideal if 0 = ab ∈ I where a, b ∈ R and a / ∈ J ( R ) imply b ∈ I. Clearly, every J -ideal is a weakly J -ideal. However, theconverse is not true in general. Indeed, (by definition) the zero ideal of any ring isweakly J -ideal but for example h ¯0 i is not a J -ideal of the ring Z . For a non-trivialexample, we present Example 1.Among many other results in this paper, in section 2, we start with a char-acterization for quasi-local rings in terms of weakly J -ideals (Theorem 1). Manyequivalent characterizations of weakly J -ideals for any commutative ring are pre-sented in Proposition 1 and Theorem 3. As a generalization of prime ideals theconcept of weakly prime ideals was first introduced in [3] by Anderson et al. Aproper ideal of a ring R is said to be a weakly prime ideal if whenever a, b ∈ R with 0 = ab ∈ I , then a ∈ I or b ∈ I . We give examples to show that weaklyprime and weakly J -ideals are not comparable. Then we justify the relationships Date : February, 2021.2020
Mathematics Subject Classification.
Key words and phrases. weakly J -ideal, J -ideal, quasi J -ideal.This paper is in final form and no version of it will be submitted for publication elsewhere. between these two concepts in Proposition 3 and Corollary 2. Further, for twoweakly J -ideals I and I of a ring R , we show that I ∩ I and I + I are weakly J -ideals of R , but I I is not so (see Propositions 4, 12 and Example 5).Recall from [4] (resp. [9]) that a ring R is called presimplifiable (resp. quasipresimplifiable) if whenever a, b ∈ R with a = ab , then a = 0 or b ∈ U ( R ) (resp. a ∈ N ( R ) or b ∈ U ( R )). It is well known from [1] that presimplifiable propertydoes not pass in general to homomorphic images. However, we show that this holdsunder a certain condition: If I a weakly J -ideal of a (quasi) presimplifiable ring R , then R/I is a (quasi) presimplifiable ring (see Corollary 5 and Proposition 9).Moreover, we investigate weakly J -ideals under various contexts of constructionssuch as direct products, localizations, homomorphic images (see Propositions 5, 6and 7).Let R be a commutative ring with identity and M an R -module. We recall that R (+) M = { ( r, m ) : r ∈ R, m ∈ M } with coordinate-wise addition and multiplica-tion defined as ( r , m )( r , m ) = ( r r , r m + r m ) is a commutative ring withidentity (1 , M . For an ideal I of R and asubmodule N of M , I (+) N is an ideal of R (+) M if and only if IM ⊆ N . More-over, the Jacobson radical of R (+) M is J ( R (+) M ) = J ( R )(+) M , [2]. We clarifythe relationships between weakly J -ideals in a ring R and in an idealization ring R (+) M in Theorem 5. The idealization can be used to extend results about idealsto modules and to provide interesting examples of commutative rings.In Section 3, for a ring R and an ideal J of R , we examine weakly J -ideals of anamalgamated ring R ⋊⋉ f J along J . Some characterizations of (weakly) J -ideals ofthe form I ⋊⋉ f J and ¯ K f of the amalgamation R ⋊⋉ f J where J ⊆ J ( S ) are given(see Theorems 6, 7 and Corollaries 6, 8). Finally, we give various counter examplesassociated with the stability of weakly J -ideals in these algebraic structures (seeExamples 2, 3, 4, 6). 2. Properties of weakly J -ideals In this section, we discuss some of the basic definitions and fundamental resultsconcerning weakly J -ideal. Among many other properties, we present a number ofcharacterizations of such class of ideals. Definition 1.
Let R be a ring. A proper ideal I of R is called a weakly J -ideal ifwhenever a, b ∈ R such that = ab ∈ I and a / ∈ J ( R ) , then b ∈ I . Clearly any J -ideal is a weakly J -ideal. The converse is not true. For a nontrivial example we have the following: Example 1.
Consider the idealization ring R = Z (+) ( Z × Z ) and consider theideal I = 0(+) h (¯1 , ¯0) i of R . Then I is not a J -ideal of R since for example, (2 , (¯0 , ¯0))(0 , (¯1 , ¯1)) = (0 , (¯0 , ¯0)) ∈ I and (2 , (¯0 , ¯0)) / ∈ J ( R ) but (0 , (¯1 , ¯1)) / ∈ I . On theother hand, I is a weakly J -ideal of R . Indeed, let ( r , (¯ a, ¯ b )) , ( r , (¯ c, ¯ d )) ∈ R suchthat (0 , (¯0 , ¯0)) = ( r , (¯ a, ¯ b ))( r , (¯ c, ¯ d )) ∈ I and ( r , (¯ a, ¯ b )) / ∈ J ( R ) . Then r = 0 and ( r r , r . (¯ c, ¯ d ) + r . (¯ a, ¯ b )) ∈ I . It follows that r r = 0 and r . (¯ c, ¯ d ) + r . (¯ a, ¯ b ) ∈h (1 , i and so r = 0 and r . (¯ c, ¯ d ) ∈ h (1 , i . By assumption, we must also have r . (¯ c, ¯ d ) = (¯0 , ¯0) . If (¯ c, ¯ d ) = (¯1 , ¯1) or (¯0 , ¯1) , then r . (¯ c, ¯ d ) ∈ h (1 , i if and only if r ∈ h i and so r . (¯ c, ¯ d ) = (¯0 , ¯0) , a contradiction. Thus, (¯ c, ¯ d ) ∈ h (¯1 , ¯0) i and I is aweakly J -ideal of R . EAKLY J-IDEALS OF COMMUTATIVE RINGS 3
However, the classes of J -ideals, quasi J -ideals and weakly J -ideals coincide inany quasi local ring. Theorem 1.
For a ring R , the following statements are equivalent: (1) R is a quasi-local ring.(2) Every proper ideal of R is a J -ideal.(3) Every proper ideal of R is a quasi J -ideal.(4) Every proper ideal of R is a weakly J -ideal.(5) Every proper principal ideal of R is a weakly J -ideal. Proof. (1) ⇔ (2) ⇔ (3) [9, Theorem 3].(2) ⇒ (4) ⇒ (5) Clear.(5) ⇒ (1) Let M be a maximal ideal of R . If M = 0, the result follows clearly.Otherwise, let 0 = a ∈ M . Now, h a i is a weakly J -ideal and 0 = a. ∈ h a i .If a / ∈ J ( R ), then 1 ∈ h a i , a contradiction. Thus, a ∈ J ( R ) and M = J ( R ).Therefore, R is quasi-local ring. (cid:3) Let I be a proper ideal of R . We denote by J ( I ), the intersection of all maximalideals of R containing I . Next, we obtain the following characterization for weakly J -ideals of R . Proposition 1.
For a proper ideal I of R , the following statements are equivalent. (1) I is a weakly J -ideal of R .(2) I ⊆ J ( R ) and whenever a, b ∈ R with 0 = ab ∈ I , then a ∈ J ( I ) or b ∈ I . Proof. (1) ⇒ (2) Suppose I is a weakly J -ideal of R . Let 0 = a ∈ I . Since 0 = a · ∈ I and 1 / ∈ I , then a ∈ J ( R ). Hence, I ⊆ J ( R ). The other claim of (2) follows clearlysince J ( R ) ⊆ J ( I ).(2) ⇒ (1) Suppose that 0 = ab ∈ I and a / ∈ J ( R ) . Since I ⊆ J ( R ), we concludethat J ( I ) ⊆ J ( J ( R )) = J ( R ) and so we get a / ∈ J ( I ). Thus, b ∈ I and I is a weakly J -ideal of R . (cid:3) We recall that a ring R is called semiprimitive if J ( R ) = 0. By (2) of Proposition1, we conclude that 0 is the only weakly J -ideal in any semiprimitive ring.Next, we show that a weakly J -ideal I that is not a J -ideal of a ring R satisfies I = 0. Theorem 2.
Let I be a weakly J -ideal of a ring R that is not a J -ideal. Then I = 0 .Proof. Suppose I = 0. We prove that I is a J -ideal. Let a, b ∈ R such that ab ∈ I and a / ∈ J ( R ). If ab = 0, then b ∈ I since I is a weakly J -ideal. Suppose ab = 0.If aI = 0, then 0 = ax for some x ∈ I and so 0 = a ( b + x ) ∈ I . Again, since I is a weakly J -ideal, b + x ∈ I and so b ∈ I . If bI = 0, then by = 0 for some y ∈ I ⊆ J ( R ). Since 0 = yb = ( y + a ) b ∈ I and clearly y + a / ∈ J ( R ), then b ∈ I .So, we may assume that aI = bI = 0. Since I = 0, then there exist x, y ∈ I suchthat yx = 0. Hence, 0 = yx = ( y + a )( x + b ) ∈ I and y + a / ∈ J ( R ) imply that x + b ∈ I . Therefore, b ∈ I and I is a J -ideal of R . (cid:3) However an ideal I satisfies I = 0 need not be a weakly J -ideal. For example,the ideal I = 0(+)4 Z of R = Z (+) Z satisfies I = 0. But, I is not a weakly J -idealof R as (0 , = (2 , , ∈ I with (2 , / ∈ J ( R ) and (0 , / ∈ I .As a corollary of Theorem 2, we have: HANI A. KHASHAN AND ECE YETKIN CELIKEL
Corollary 1.
Let I be a weakly J -ideal of a ring R that is not a J -ideal. Then (1) I ⊆ N ( R ) . (2) Whenever M is an R -module and IM = M , then M = 0 . In particular, if R is a reduced ring, then by Corollary 1, a non-zero proper ideal I is a weakly J -ideal if and only if I is a J -ideal.In the following theorem, we give some other characterizations of weakly J -ideals. Theorem 3.
Let I be a proper ideal of a ring R. Then the following are equivalent: (1) I is a weakly J -ideal of R .(2) ( I : a ) = I ∪ (0 : a ) for every a ∈ R \ J ( R ).(3) ( I : a ) ⊆ J ( R ) ∪ (0 : a ) for every a ∈ R \ I .(4) If a ∈ R and K is an ideal of R with 0 = Ka ⊆ I , then K ⊆ J ( R ) or a ∈ I .(5) If K and L are ideals of R with 0 = KL ⊆ I , then K ⊆ J ( R ) or L ⊆ I. Proof. (1) ⇒ (2) Let a ∈ R \ J ( R ). It is clear that I ∪ (0 : a ) ⊆ ( I : a ). Let x ∈ ( I : a )so that ax ∈ I . If ax = 0, then x ∈ I as I is a weakly J -ideal. If ax = 0, then x ∈ (0 : a ). Thus, ( I : a ) ⊆ I ∪ (0 : a ) and the equality holds.(2) ⇒ (1) Let a, b ∈ R such that 0 = ab ∈ I and a / ∈ J ( R ). Then b / ∈ (0 : a ) andso b ∈ ( I : a ) ⊆ I .(1) ⇒ (3) Similar to the proof of (1) ⇒ (2).(3) ⇒ (4) Suppose that 0 = aK ⊆ I and a / ∈ I . Then ( I : a ) = (0 : a ) and so K ⊆ ( I : a ) ⊆ J ( R ), as needed.(4) ⇒ (5) Assume on the contrary that there are ideals K and L of R such that0 = KL ⊆ I but K * J ( R ) and L * I . Since KL = 0, there exists a ∈ L suchthat 0 = Ka ⊆ I . Since K * J ( R ), we have a ∈ I by (4). Now, choose anelement x ∈ L \ I . Similar to the previous argument, we conclude Kx = 0. (Indeed,if Kx = 0, then x ∈ I ). Hence, 0 = K ( a + x ) ⊆ I and K * J ( R ) imply that( a + x ) ∈ I . Thus, x ∈ I , a contradiction.(5) ⇒ (1) Let a, b ∈ R with 0 = ab ∈ I . Write K = < a > and L = h b i . Then theresult follows directly by (5). (cid:3) Proposition 2.
Let S be a non-empty subset of R . If I and (0 : S ) are weakly J -ideals of R where S * I , then so is ( I : S ) .Proof. We first note that ( I : S ) is proper in R since otherwise, S ⊆ I , a contradic-tion. Let 0 = ab ∈ ( I : S ) and a / ∈ J ( R ) . If abS = 0, then bS ⊆ I and so b ∈ ( I : S )as I is weakly J -ideal. If abS = 0, then 0 = ab ∈ (0 : S ) which implies b ∈ (0 : S )as (0 : S ) is a weakly J -ideal of R . Thus, again b ∈ ( I : S ) as required. (cid:3) Recall that a proper ideal P of a ring R is called weakly prime if whenever a, b ∈ R such that 0 = ab ∈ P , then a ∈ P or b ∈ P . In general, weakly J -idealsand weakly prime ideals are not comparable. For example, any non-zero primeideal in the domain of integers is weakly prime that is not a weakly J -ideal. On theother hand, the ideal h i is a weakly J -ideal in the ring Z which is clearly notweakly prime. However, for ideals contained in the Jacobson radical, we clarify inthe following proposition that weakly prime ideals are weakly J-ideals. The proofis straightforward. Proposition 3. If I is a weakly prime ideal of a ring R and I ⊆ J ( R ) , then I isa weakly J -ideal. EAKLY J-IDEALS OF COMMUTATIVE RINGS 5
The converse of the previous proposition holds under certain conditions:
Corollary 2.
Let I be an ideal of a ring R . Suppose I is maximal with respect tothe property: I and (0 : a ) are weakly J -ideals for all a / ∈ I . Then I is weakly primein R .Proof. Let a, b ∈ R such that 0 = ab ∈ I and a / ∈ I . If we choose S = { a } inProposition 2, then we conclude that ( I : a ) is a weakly J -ideal of R . Moreover,clearly c / ∈ I for all c / ∈ ( I : a ) and so (0 : c ) is a weakly J -ideals. By maximality of I , we must have b ∈ ( I : a ) = I as required. (cid:3) Proposition 4. If { I i : i ∈ ∆ } is a non empty family of weakly J -ideals of a ring R , then T i ∈ ∆ I i is a weakly J -ideal.Proof. Let a, b ∈ R such that 0 = ab ∈ T i ∈ ∆ I i and a / ∈ J ( R ). Since for all i ∈ ∆, I i isa weakly J -ideal of R , we have b ∈ I i . Hence, b ∈ T i ∈ ∆ I i and the result follows. (cid:3) In general, the converse of Proposition 4 is not true. For example, while h ¯0 i = h ¯2 i∩h ¯3 i is a weakly J -ideal of Z , non of the ideals h ¯2 i and h ¯3 i are (weakly) J -ideals.Next, we characterize weakly J -ideals of a Cartesian product of two rings. Proposition 5.
Let R = R × R be a decomposable ring and I be a non-zeroproper ideal of R . Then the following statements are equivalent. (1) I is a weakly J -ideal of R. (2) I = I × R where I is a J -ideal of R or I = R × I where I is a J -idealof R . (3) I is a J -ideal of R. Proof. (1) ⇒ (2) Let I = I × I be a non-zero weakly J -ideal of R . Assume I and I are proper in R and R , respectively and choose 0 = ( a, b ) ∈ I . Then0 = ( a, , b ) ∈ I and neither ( a, ∈ J ( R ) nor (1 , b ) ∈ I , a contradiction. Wemay assume with no loss of generality that I = R and I = R . Since clearly I = 0, I is a J -ideal of R by Corollary 1 (2). Let a, b ∈ R such that ab ∈ I and a / ∈ J ( R ). Then ( a, b, ∈ I and ( a, / ∈ J ( R ) imply that ( b, ∈ I and so b ∈ I as required.(2) ⇒ (3) We may assume that I = I × R where I is a J -ideal of R . Supposethat ( a, b )( c, d ) ∈ I and ( a, b ) / ∈ J ( R ) . Then clearly a / ∈ J ( R ) and ac ∈ I whichimply c ∈ I . Thus, ( c, d ) ∈ I and we are done.(3) ⇒ (1) straightforward. (cid:3) Corollary 3.
Let R = R × R be a decomposable ring. If I is a weakly J -ideal of R that is not a J -ideal, then I = 0 . Let I be a proper ideal of R. In the following, the notation Z I ( R ) denotes theset of { r ∈ R | rs ∈ I for some s ∈ R \ I } . Proposition 6.
Let S be a multiplicatively closed subset of a ring R such that J ( S − R ) = S − J ( R ) . (1) If I is a weakly J -ideal of R such that I ∩ S = ∅ , then S − I is a weakly J -ideal of S − R .(2) If S − I is a weakly J -ideal of S − R and S ∩ Z ( R ) = S ∩ Z I ( R ) = S ∩ Z J ( R ) ( R ) = ∅ , then I is a weakly J -ideal of R . HANI A. KHASHAN AND ECE YETKIN CELIKEL
Proof. (1) Let 0 = as bs ∈ S − I and as / ∈ J ( S − R ) for some as , bs ∈ S − R. Then0 = uab ∈ I for some u ∈ S . Since clearly a / ∈ J ( R ) and I is a weakly J -ideal, wehave ub ∈ I . Hence bs = ubus ∈ S − I , as needed.(2) Let a, b ∈ R and 0 = ab ∈ I . Then a b ∈ S − I . If a b = 0, then uab = 0 forsome u ∈ S. Since S ∩ Z ( R ) = ∅ , we have ab = 0, a contradiction. Since S − I is aweakly J -ideal of S − R , 0 = a b ∈ S − I implies either a ∈ J ( S − R ) = S − J ( R )or b ∈ S − I . If a ∈ S − J ( R ), then there exists u ∈ S with ua ∈ J ( R ). Since S ∩ Z J ( R ) ( R ) = ∅ , then a ∈ J ( R ). If b ∈ S − I , then there exists v ∈ S with vb ∈ I and so b ∈ I as S ∩ Z I ( R ) = ∅ . Therefore, I is a weakly J -ideal of R . (cid:3) Proposition 7.
Let f : R → R be a ring homomorphism. Then the followingstatements hold. (1) If f is a monomorphism and I is a weakly J -ideal of R , then f − ( I ) isa weakly J -ideal of R . (2) If f is an epimorphism and I is a weakly J -ideal of R with Ker ( f ) ⊆ I ,then f ( I ) is a weakly J -ideal of R . Proof. (1) Suppose that a, b ∈ R with 0 = ab ∈ f − ( I ) and a / ∈ J ( R ). Observethat f ( a ) / ∈ J ( R ) by the proof of [8, Proposition 2.23 (2)]. Since Ker ( f ) = 0,we have 0 = f ( ab ) = f ( a ) f ( b ) ∈ I . Since I is a weakly J -ideal of R , we get f ( b ) ∈ I , and so b ∈ f − ( I ).(2) Let a, b ∈ R and 0 = ab ∈ f ( I ). Since f is onto, a = f ( x ) and b = f ( y )for some x, y ∈ R . Hence, 0 = f ( x ) f ( y ) = f ( xy ) ∈ f ( I ). Since Ker ( f ) ⊆ I , wehave 0 = xy ∈ I which implies x ∈ J ( R ) or y ∈ I . Thus, a = f ( x ) ∈ J ( R ) by[8, Lemma 2.22] or b = f ( y ) ∈ f ( I ) and we are done. (cid:3) Corollary 4.
Let I and K be proper ideals of R with K ⊆ I . (1) If I is a weakly J -ideal of R , then I/K is a weakly J -ideal of R/K .(2) If K is a J -ideal of R and I/K is a weakly J -ideal of R/K , then I is a J -ideal of R .(3) If K is a weakly J -ideal of R and I/K is a weakly J -ideal of R/I , then I is a weakly J -ideal of R . Proof. (1) Follows by Proposition 7.(2) Let a, b ∈ R with ab ∈ I and a / ∈ J ( R ). If ab ∈ K , then b ∈ K ⊆ I .Now, suppose that ab / ∈ K . Since K is a J -ideal, K ⊆ J ( R ) and so clearly a + K / ∈ J ( R/K ). Since K = ( a + K )( b + K ) = ab + K ∈ I/K and
I/K is weakly J -ideal, we have ( b + K ) ∈ I/K . Thus, b ∈ I and we are done.(3) Similar to (2). (cid:3) Recall from [4] that a ring R is called presimplifiable if whenever a, b ∈ R with a = ab , then a = 0 or b ∈ U ( R ). Equivalently, R is presimplifiable if and only if Z ( R ) ⊆ J ( R ). The next result states that in a presimplifiable ring, weakly J -idealsand J -ideals coincide. Proposition 8.
Every weakly J -ideal of a presimplifiable ring is a J -ideal.Proof. Let R be a presimplifiable ring and I be a weakly J -ideal of R. Then 0 is a J -ideal by [9, Corollary 5]. So, the claim follows from Corollary 4 (2). (cid:3) EAKLY J-IDEALS OF COMMUTATIVE RINGS 7
It is well known that presimplifiable property does not pass in general to homo-morphic images, [1]. In view of Proposition 8 and [9, Theorem 8], we prove thatthis holds under a certain condition.
Corollary 5. If R is a presimplifiable ring and I is a weakly J -ideal of R , then R/I is presimplifiable.
Recall from [9] that a proper ideal I of a ring R is said to be quasi J -ideal if √ I is a J -ideal of R . A ring R is called quasi presimplifiable if whenever a, b ∈ R with a = ab , then a ∈ N ( R ) or b ∈ U ( R ). We need the following lemma which justifiesthe relation between these two concepts. Lemma 1. [9]
Let I be a proper ideal of a ring R . Then I is a quasi J -ideal of R if and only if I ⊆ J ( R ) and R/I is quasi presimplifiable.
Proposition 9.
Let R be a quasi presimplifiable ring and I a weakly J -ideal of R .Then R/I is a quasi presimplifiable ring.Proof.
Suppose ab ∈ √ I and a / ∈ J ( R ). Then a n b n ∈ I for some n ∈ N . Suppose a n b n = 0. Since R is quasi presimplifiable, then 0 is a quasi J -ideal of R by Lemma1 and so N ( R ) is a J -ideal. Hence, ab ∈ N ( R ) implies b ∈ N ( R ) ⊆ √ I . Now,suppose that 0 = a n b n ∈ I . Since clearly a n / ∈ J ( R ) , we conclude that b n ∈ I and b ∈ √ I . Hence √ I is a J -ideal and so I is a quasi J -ideal of R . Therefore, R/I isa quasi presimplifiable ring by Lemma 1. (cid:3)
Proposition 10.
Let R be a Noetherian domain and I be a proper ideal of R .Then I is a J -ideal of R if and only if I ⊆ J ( R ) and I/I n is a weakly J -ideal of R/I n for all positive integers n. Proof.
Suppose I is a J -ideal of R . Then I ⊆ J ( R ) by Proposition 1 and I/I n is a weakly J -ideal of R/I n by Corollary 4 (1). Conversely, suppose that for all n ∈ N , I/I n is a weakly J -ideal of R/I n and let ab ∈ I . If ab / ∈ I n for some n ≥ I n = ( a + I n )( b + I n ) ∈ I/I n which implies ( a + I n ) ∈ J ( R/I n ) or( b + I n ) ∈ I/I n . Since by assumption, I n ⊆ I ⊆ J ( R ), then J ( R/I n ) = J ( R ) /I n .Thus, a ∈ J ( R ) or b ∈ I , as needed. Now, assume that ab ∈ I n for all n. Since R is Noetherian, the Krull’s intersection theorem implies that ∞ T n =1 I n = 0. Therefore, ab = 0 and since R is a domain, we conclude a = 0 or b = 0 and we are done. (cid:3) Definition 2.
Let I be a non-zero ideal of a ring R . An element a + I ∈ R/I is called a strong zero divisor in
R/I if there exists I = b + I ∈ R/I such that ( a + I )( b + I ) = I and ab = 0 . It is clear that any strong zero divisor in
R/I is a zero divisor. The converse isnot true since for example ¯2 + h ¯4 i is a zero divisor in Z / h ¯4 i which is not a strongzero divisor.For an ideal I of a ring R , we denote the set of strong zero divisors of R/I by SZ ( R/I ). It is clear that if I = 0, (e.g. R is a field), then SZ ( R/I ) = φ .Let I be a non-zero ideal of a ring R . Analogous to to the presimplifiable rings,we define a ring R/I to be S -presimplifiable if SZ ( R/I ) ⊆ J ( R/I ). Next, wecharacterize non-zero weakly J -ideals in terms of S -presimplifiable quotient rings. Theorem 4.
Let I be a non-zero ideal of a ring R . Then I is a weakly J -ideal of R if and only if I ⊆ J ( R ) and R/I is S -presimplifiable. HANI A. KHASHAN AND ECE YETKIN CELIKEL
Proof.
Suppose I is a weakly J -ideal of R and note that I ⊆ J ( R ) by Proposition1. Let a + I ∈ SZ ( R/I ) and choose I = b + I ∈ R/I such that ( a + I )( b + I ) = I and ab = 0. Then 0 = ab ∈ I and b / ∈ I . Hence, a ∈ J ( R ) as I is a weakly J -ideal.Therefore, ( a + I ) ∈ J ( R ) /I = J ( R/I ) and we are done. Conversely, let a, b ∈ R such that 0 = ab ∈ I and b / ∈ I . Then clearly, a + I is a strong zero divisor in R/I and so a + I ∈ J ( R/I ). It follows that a ∈ J ( R ) and so I is a weakly J -ideal of R . (cid:3) It is clear that for a non zero ideal I of a ring R , if R/I is presimplifiable, thenit is S -presimplifiable. However, we have seen in Example 1 that 0(+) h (¯1 , ¯0) i isa weakly J -ideal of Z (+) ( Z × Z ) that is not a J -ideal. In view of the abovetheorem and [9, Theorem 8], we conclude that Z (+) ( Z × Z ) / h (¯1 , ¯0) i is an S -presimplifiable ring that is not presimplifiable.It is well known that for any ring R , J ( R [ | x | ]) = J ( R ) + xR [ | x | ]. Proposition 11.
Let R be a ring. If I is a weakly J -ideal of R [ | x | ] (resp., R [ x ] ),then I ∩ R is a weakly J -ideal of R .Proof. (1) Suppose I is a weakly J -ideal of R [ | x | ]. Let 0 = ab ∈ I ∩ R and a / ∈ J ( R )for a, b ∈ R . Then 0 = ab ∈ I and a / ∈ J ( R [ | x | ]) imply that b ∈ I . Thus, b ∈ I ∩ R as needed. (cid:3) A proper ideal I in a ring R is called superfluous if whenever J is an ideal of R with I + J = R , then J = R . Lemma 2.
If an ideal I of a ring R is a weakly J -ideal, then it is superfluous.Proof. Suppose I + J = R for some ideal J of R . Then 1 = x + y for some x ∈ I and y ∈ J and so 1 − y ∈ I ⊆ J ( R ) by Proposition 1. Thus y ∈ J is a unit and J = R . (cid:3) Proposition 12.
Let I and I be weakly J -ideals of a ring R . Then I + I is aweakly J -ideal of R .Proof. Suppose that I and I are weakly J -ideals. Then I + I is proper by Lemma2. Since I ∩ I is a weakly J -ideal by Proposition 4, then I / ( I ∩ I ) is a weakly J -ideal of R/ ( I ∩ I ) by Corollary 4 (1). From the isomorphism I / ( I ∩ I ) ∼ =( I + I ) /I , we conclude that ( I + I ) /I is a weakly J -ideal of R/I . Thus, I + I is a weakly J -ideal of R by Corollary 4 (3). (cid:3) Next, we generalize the concept of J -multiplicatively closed subset of a ring R ,[8, Definition 2.27]. Definition 3.
Let S be a non empty subset of a ring R such that R − J ( R ) ⊆ S .Then S is called weakly J -multiplicatively closed if ab ∈ S or ab = 0 for all a ∈ R − J ( R ) and b ∈ S . Similar to the relation between J -ideals and J -multiplicatively closed subsets ofrings, we have: Proposition 13.
An ideal I is a weakly J -ideal of a ring R if and only if R − I isa weakly J -multiplicatively closed subset of R . EAKLY J-IDEALS OF COMMUTATIVE RINGS 9
Proof. If I is a weakly J -ideal of R , then I ⊆ J ( R ) and so R − J ( R ) ⊆ R − I .Let a ∈ R − J ( R ) and b ∈ R − I . If ab = 0, then we are done. Otherwise,suppose ab = 0. Since I is a weakly J -ideal, then ab ∈ R − I and so R − I isa weakly J -multiplicatively closed subset of R . Conversely, suppose R − I is a J -multiplicatively closed subset of R . Let a, b ∈ R such that 0 = ab ∈ I and a / ∈ J ( R ).If b ∈ R − I , Then ab ∈ R − I as R − I is a weakly J -multiplicatively closed subset.This contradiction implies b ∈ I and so I is a weakly J -ideal of R . (cid:3) Proposition 14.
Let S be a weakly J -multiplicatively closed subset of a ring R such that S ∩ S a/ ∈ J ( R ) (0 : a ) = φ . If an ideal I of R is maximal with respect to theproperty I ∩ S = φ , then I is a weakly J -ideal of a ring R .Proof. Suppose I is not a weakly J -ideal of R . Then there are a / ∈ J ( R ) and b / ∈ I such that ab ∈ I but ab = 0. Since I ( I : a ), then ( I : a ) ∩ S = φ and so thereexists s ∈ ( I : a ) ∩ S . Now, as ∈ I and since S is weakly J -multiplicatively closed,we have either as ∈ S or as = 0. If as ∈ S , then S ∩ I = φ , a contradiction. If as = 0, then s ∈ S ∩ S a/ ∈ J ( R ) (0 : a ) which is also a contradiction. Therefore, I is aweakly J -ideal of a ring R . (cid:3) Next, we justify the relation between weakly J -ideals of a ring R and those ofthe idealization ring R (+) M . Theorem 5.
Let I be an ideal of a ring R and N be a submodule of an R -module M . (1) If I (+) N is a weakly J -ideal of the idealization ring R (+) M , then I is aweakly J -ideal of R . (2) I (+) M is a weakly J -ideal of R (+) M if and only if I is a weakly J -ideal of R and for x, y ∈ R with xy = 0 but x / ∈ J ( R ) and y / ∈ I , x, y ∈ Ann ( M ) .Proof. (1) If I = R , then clearly I (+) N = R (+) M , a contradiction. Let a, b ∈ R with 0 = ab ∈ I and a / ∈ J ( R ). Then (0 , = ( a, b, ∈ I (+) N and ( a, / ∈ J ( R (+) M ) = J ( R )(+) M . Since I (+) N is a weakly J -ideal, we have ( b, ∈ I (+) N and b ∈ I as needed.(2) Suppose I (+) M is a weakly J -ideal. Then I is so by (1). Now, for x, y ∈ R ,suppose xy = 0 but x / ∈ J ( R ) and y / ∈ I . If x / ∈ Ann ( M ), then there exists m ∈ M such that xm = 0. Hence, (0 , = ( x, y, m ) ∈ I (+) M but ( x, / ∈ J ( R (+) M )and ( y, m ) / ∈ I (+) M , a contradiction. Therefore, x ∈ Ann ( M ). Similarly, we canprove that y ∈ Ann ( M ). Conversely, suppose (0 , = ( a, m ) ( b, m ) ∈ I (+) M and( a, m ) / ∈ J ( R (+) M ) for ( a, m ) , ( b, m ) ∈ R (+) M . Then ab ∈ I and a / ∈ J ( R ).If ab = 0, then b ∈ I as I is a weakly J -ideal and so ( b, m ) ∈ I (+) M . Suppose ab = 0 but neither a ∈ J ( R ) nor b ∈ I . By assumption, a, b ∈ Ann ( M ) and so( a, m ) ( b, m ) = (0 , (cid:3) However, in general, if I is a weakly J -ideal of R , then I (+) M need not be so.For example, although 0 is a (weakly) J -ideal of Z , the ideal 0(+) h i of Z (+) Z isnot a weakly J -ideal. Indeed, (0 , = (2 , , ∈ h i but (2 , / ∈ J ( Z (+) Z )and (0 , / ∈ h i . (Weakly) J -ideals of Amalgamated Rings Along an Ideal Let R and S be two rings, J be an ideal of S and f : R → S be a ring homomor-phism. The set R ⋊⋉ f J = { ( r, f ( r ) + j ) : r ∈ R , j ∈ J } is a subring of R × S (withidentity element (1 R , S ) ) called the amalgamation of R and S along J with re-spect to f . In particular, if Id R : R → R is the identity homomorphism on R , then R ⋊⋉ J = R ⋊⋉ Id R J = { ( r, r + j ) : r ∈ R , j ∈ J } is the amalgamated duplication ofa ring along an ideal J . This construction has been first defined and studied byD’Anna and Fontana, [5]. Many properties of this ring have been investigated andanalyzed over the last two decades, see for example [6], [7].Let I be an ideal of R and K be an ideal of f ( R ) + J . Then I ⋊⋉ f J = { ( i, f ( i ) + j ) : i ∈ I , j ∈ J } and ¯ K f = { ( a, f ( a ) + j ) : a ∈ R , j ∈ J , f ( a ) + j ∈ K } are ideals of R ⋊⋉ f J , [7]. Lemma 3. [7]
Let R , S , J and f be as above. The set of all maximal ideals of R ⋊⋉ f J is M ax ( R ⋊⋉ f J ) = (cid:8) M ⋊⋉ f J : M ∈ M ax ( R ) (cid:9) ∪ (cid:8) ¯ Q f : Q ∈ M ax ( S ) \ V ( J ) (cid:9) where V ( J ) denotes the set of all prime ideals containing J . In particular if J ⊆ J ( S ) (e.g. S is quasi-local or J is a weakly J -ideal), then weconclude from Lemma 3 that J ( R ⋊⋉ f J ) = J ( R ) ⋊⋉ f J . The proof of the followingproposition is straightforward by using Theorem 1. Proposition 15.
Consider the amalgamation of rings R and S along the ideal J of S with respect to a homomorphism f . If R is a quasi-local ring and J ⊆ J ( S ) ,then every ideal of R ⋊⋉ f J is a (weakly) J -ideal. Next, we give a characterization of (weakly) J -ideals of the form I ⋊⋉ f J and ¯ K f of the amalgamation R ⋊⋉ f J when J ⊆ J ( S ). Theorem 6.
Consider the amalgamation of rings R and S along the ideals J of S with respect to a homomorphism f . Let I be an ideal of R . Then (1) If I ⋊⋉ f J is a J -ideal of R ⋊⋉ f J , then I is a J -ideal of R . Moreover, theconverse is true if J ⊆ J ( S ) . (2) If I ⋊⋉ f J is a weakly J -ideal of R ⋊⋉ f J , then I is a weakly J -ideal of R andfor a, b ∈ R with ab = 0 , but a / ∈ J ( R ) , b / ∈ I , then f ( a ) j + f ( b ) i + ij = 0 for every i, j ∈ J . Moreover, the converse is true if J ⊆ J ( S ) .Proof. (1) Suppose I ⋊⋉ f J is a J -ideal of R ⋊⋉ f J . Let a, b ∈ R such that ab ∈ I and a / ∈ J ( R ). Then ( a, f ( a ))( b, f ( b )) ∈ I ⋊⋉ f J and ( a, f ( a )) / ∈ J ( R ⋊⋉ f J ) sinceotherwise a ∈ M for each M ∈ M ax ( R ) by Lemma 3, a contradiction. It followsthat ( b, f ( b )) ∈ I ⋊⋉ f J and so b ∈ I as needed.Moreover, suppose J ⊆ J ( S ) and I is a a J -ideal of R . Let ( a, f ( a )+ j )( b, f ( b )+ j ) = ( ab, f ( ab ) + f ( a ) j + f ( b ) j + j j ) ∈ I ⋊⋉ f J for ( a, f ( a ) + j ) , ( b, f ( b ) + j ) ∈ R ⋊⋉ f J . If ( a, f ( a ) + j ) / ∈ J ( R ⋊⋉ f J ) = J ( R ) ⋊⋉ f J , then a / ∈ J ( R ). Since ab ∈ I ,we conclude that b ∈ I and so ( b, f ( b ) + j ) ∈ I ⋊⋉ f J . Thus, I ⋊⋉ f J is a J -ideal of R ⋊⋉ f J .(2) Suppose I ⋊⋉ f J is a weakly J -ideal of R ⋊⋉ f J and let a, b ∈ R such that0 = ab ∈ I and a / ∈ J ( R ). Then (0 , = ( a, f ( a ))( b, f ( b )) ∈ I ⋊⋉ f J and ( a, f ( a )) / ∈ J ( R ⋊⋉ f J ) by Lemma 3. It follows that ( b, f ( b )) ∈ I ⋊⋉ f J and so b ∈ I . For thesecond claim, suppose there exist i, j ∈ J such whenever a, b ∈ R with ab = 0, but a / ∈ J ( R ), b / ∈ I , f ( a ) j + f ( b ) i + ij = 0. Then (0 , = ( a, f ( a ) + i )( b, f ( b ) + j ) =( ab, f ( ab ) + f ( a ) j + f ( b ) i + ij ) = (0 , f ( a ) j + f ( b ) i + ij ) ∈ I ⋊⋉ f J . This is a EAKLY J-IDEALS OF COMMUTATIVE RINGS 11 contradiction since I ⋊⋉ f J is a weakly J -ideal, ( a, f ( a ) + i ) / ∈ J ( R ⋊⋉ f J ) and( b, f ( b )+ j ) / ∈ I ⋊⋉ f J . Now, we prove the converse under the assumption J ⊆ J ( S ).Let 0 = ( a, f ( a ) + j )( b, f ( b ) + j ) = ( ab, f ( ab ) + f ( a ) j + f ( b ) j + j j ) ∈ I ⋊⋉ f J where ( a, f ( a ) + j ) / ∈ J ( R ⋊⋉ f J ) = J ( R ) ⋊⋉ f J . Then ab ∈ I and we have twocases: Case I: If ab = 0, then as clearly a / ∈ J ( R ), we have b ∈ I . Therefore, ( b, f ( b ) + j ) ∈ I ⋊⋉ f J and I ⋊⋉ f J is a weakly J -ideal of R ⋊⋉ f J . Case II:
Suppose ab = 0. If a / ∈ J ( R ) and b / ∈ I , then by assumption, f ( a ) j + f ( b ) i + ij = 0 for every i, j ∈ J . This implies ( a, f ( a ) + j )( b, f ( b ) + j ) = (0 , a ∈ J ( R ) or b ∈ I and so ( a, f ( a ) + j ) ∈ J ( R ⋊⋉ f J ) or( b, f ( b ) + j ) ∈ I ⋊⋉ f J as required. (cid:3) Corollary 6.
Consider the amalgamation of rings R and S along the ideal J ⊆ J ( S ) of S with respect to a homomorphism f . The J -ideals of R ⋊⋉ f J containing { } × J are of the form I ⋊⋉ f J where I is a J -ideal of R. Proof.
First, we note that I ⋊⋉ f J is a J -ideal of R ⋊⋉ f J for any J -ideal I of R by Theorem 6. Let K be a J -ideal of R ⋊⋉ f J containing { } × J. Consider thesurjective homomorphism ϕ : R ⋊⋉ f J → R defined by ϕ ( a, f ( a ) + j ) = a for all( a, f ( a ) + j ) ∈ R ⋊⋉ f J . Since Ker ( ϕ ) = { } × J ⊆ K , then I := ϕ ( K ) is a J -idealof R by [8, Proposition 2.23 (1)] . Since { } × J ⊆ K , we conclude that K is of theform I ⋊⋉ f J . (cid:3) As a particular case of Theorem 6, we have the following immediate corollary.
Corollary 7.
Let R be a ring and I , J be be proper ideals of R . Then (1) If I ⋊⋉ J is a J -ideal of R ⋊⋉ J , then I is a J -ideal of R . Moreover, theconverse is true if J ⊆ J ( R ) . (2) If I ⋊⋉ J is a weakly J -ideal of R ⋊⋉ J , then I is a weakly J -ideal of R andfor a, b ∈ R with ab = 0 , but a / ∈ J ( R ) , b / ∈ I , then aj + bi + ij = 0 forevery i, j ∈ J . Moreover, the converse is true if J ⊆ J ( R ) . Theorem 7.
Consider the amalgamation of rings R and S along the maximalideal J of S with respect to an epimorphism f . Let K be an ideal of S . (1) If ¯ K f is a J -ideal of R ⋊⋉ f J , then K is a a J -ideal of S . The converse istrue if f ( J ( R )) = J ( S ) + J and Ker ( f ) ⊆ J ( R ) . (2) If ¯ K f is a weakly J -ideal of R ⋊⋉ f J , then K is a weakly J -ideal of S and when f ( a ) + j / ∈ J ( S ) , f ( b ) + k / ∈ K with a, b ∈ R , j, k ∈ J and ( f ( a ) + j )( f ( b ) + k ) = 0 , then ab = 0 . The converse is true if f ( J ( R )) = J ( S ) + J and Ker ( f ) ⊆ J ( R ) .Proof. (1) Suppose ¯ K f is a J -ideal of R ⋊⋉ f J . Let x, y ∈ S , say, x = f ( a ) and y = f ( b ) for a, b ∈ R . Suppose xy ∈ K and x / ∈ J ( S ). Then ( a, f ( a )) , ( b, f ( b )) ∈ R ⋊⋉ f J such that ( a, f ( a ))( b, f ( b )) = ( ab, ( f ( a ))( f ( b ))) ∈ ¯ K f . If ( a, f ( a )) ∈ J ( R ⋊⋉ f J ),then ( a, f ( a )) ∈ ¯ Q f for all Q ∈ M ax ( S ) \ V ( J ). Moreover, since J is maximal in S , then f − ( J ) is maximal in R and so ( a, f ( a )) ∈ f − ( J ) ⋊⋉ f J . Thus, f ( a ) ∈ J = V ( J ) and so f ( a ) ∈ Q for all Q ∈ M ax ( S ), a contradiction. Therefore,( a, f ( a )) / ∈ J ( R ⋊⋉ f J ) and so ( b, f ( b )) ∈ ¯ K f as ¯ K f is a J -ideal of R ⋊⋉ f J . Hence, y = f ( b ) ∈ K and we are done.Now, suppose f ( J ( R )) = J ( S ) + J , Ker ( f ) ⊆ J ( R ) and K is a J -ideal of S .Let ( a, f ( a ) + j ) , ( b, f ( b ) + k ) ∈ R ⋊⋉ f J such that ( a, f ( a ) + j )( b, f ( b ) + k ) = ( ab, ( f ( a ) + j )( f ( b ) + k )) ∈ ¯ K f and ( a, f ( a ) + j ) / ∈ J ( R ⋊⋉ f J ). We claim that f ( a )+ j / ∈ J ( S ). Suppose not. Then f ( a ) ∈ J ( S )+ J = f ( J ( R )) and so a ∈ J ( R ) as Ker ( f ) ⊆ J ( R ). Thus, ( a, f ( a )+ j ) ∈ (cid:8) M ⋊⋉ f J : M ∈ M ax ( R ) (cid:9) . Moreover, f ( a )+ j ∈ Q for all Q ∈ M ax ( S ) implies that ( a, f ( a ) + j ) ∈ (cid:8) ¯ Q f : Q ∈ M ax ( S ) \ V ( J ) (cid:9) .It follows by Lemma 3 that ( a, f ( a ) + j ) ∈ J ( R ⋊⋉ f J ), a contradiction. Since( f ( a ) + j )( f ( b ) + k ) ∈ K and K is a J -ideal of S , then f ( b ) + k ∈ K . Hence,( b, f ( b ) + k ) ∈ ¯ K f and the result follows.(2) Suppose ¯ K f is a weakly J -ideal of R ⋊⋉ f J . Let x = f ( a ) , y = f ( b ) ∈ S for a, b ∈ R such that 0 = xy ∈ K and x / ∈ J ( S ). Then (0 , = ( a, f ( a ))( b, f ( b )) =( ab, f ( ab )) and similar to the proof of (1), we have ( a, f ( a )) / ∈ J ( R ⋊⋉ f J ). Byassumption, we have ( b, f ( b )) ∈ ¯ K f and so y = f ( b ) ∈ K . Moreover, let a, b ∈ R and j, k ∈ J such that f ( a ) + j / ∈ J ( S ) , f ( b ) + k / ∈ K and ( f ( a ) + j )( f ( b ) + k ) = 0.Suppose ab = 0. Then (0 , = ( a, f ( a ) + j )( b, f ( b ) + k ) = ( ab, ∈ ¯ K f . Similar tothe proof of (1), we conclude that ( a, f ( a ) + j ) / ∈ J ( R ⋊⋉ f J ) and ( b, f ( b ) + k ) / ∈ ¯ K f which contradict the assumption that ¯ K f is a weakly J -ideal of R ⋊⋉ f J . Thus, ab = 0 as needed.Now, we assume f ( J ( R )) = J ( S ) + J , Ker ( f ) ⊆ J ( R ) and K a weakly J -ideal of S . Let ( a, f ( a ) + j ) , ( b, f ( b ) + k ) ∈ R ⋊⋉ f J such that (0 , = ( a, f ( a ) + j )( b, f ( b ) + k ) = ( ab, ( f ( a ) + j )( f ( b ) + k )) ∈ ¯ K f and ( a, f ( a ) + j ) / ∈ J ( R ⋊⋉ f J ).Then f ( a ) + j / ∈ J ( S ) as in the proof of (1) and ( f ( a ) + j )( f ( b ) + k ) ∈ K . We havetwo cases: Case I: ( f ( a )+ j )( f ( b )+ k ) = 0. In this case we conclude directly that ( f ( b )+ k ) ∈ K . Thus, ( b, f ( b ) + k ) ∈ ¯ K f and ¯ K f is a weakly J -ideal of R ⋊⋉ f J . Case II: ( f ( a ) + j )( f ( b ) + k ) = 0. If f ( a ) + j / ∈ J ( S ) and f ( b ) + k / ∈ K , then byassumption we should have ab = 0. It follows that ( a, f ( a ) + j )( b, f ( b ) + k ) = (0 , K f is a weakly J -ideal of R ⋊⋉ f J . (cid:3) Corollary 8.
Let R be a ring, K a proper ideal of R and J a maximal ideal of R .Then (1) If ¯ K = { ( a, a + j ) : a ∈ R , j ∈ J , a + j ∈ K } is a J -ideal of R ⋊⋉ J , then K is a J -ideal of R . Moreover, the converse is true if J ⊆ J ( R ) . (2) If ¯ K is a weakly J -ideal of R ⋊⋉ J , then I is a weakly J -ideal of R and when a + j / ∈ J ( R ) , b + k / ∈ K with a, b ∈ R , j, k ∈ J and ab + ak + bj + jk = 0 ,then ab = 0 . Moreover, the converse is true if J ⊆ J ( R ) . In the following example, we prove that the condition J ⊆ J ( S ) can not bediscarded in the proof of the converses of (1) and (2) in Theorem 6. Example 2.
Let R = Z (+) Z , I = 0(+) Z and J = h i (+) Z * J ( R ) . Then I is a weakly J -ideal of R by Theorem 6 (2). Moreover, one can easily see that thereare no ( r , m ) , ( r , m ) ∈ R with ( r , m )( r , m ) = (0 , ¯0) , but ( r , m ) / ∈ J ( R ) , ( r , m ) / ∈ I . Now, ((0 , ¯1) , (2 , ¯1)) , ((1 , ¯0) , (1 , ¯0) ∈ R ⋊⋉ J with ((0 , ¯1) , (2 , ¯1))((1 , ¯0) , (1 , ¯0)) =((0 , ¯1) , (2 , ¯1)) ∈ I ⋊⋉ J \ ((0 , ¯0) , (0 , ¯0)) . Moreover, ((0 , ¯1) , (2 , ¯1)) / ∈ J ( R ⋊⋉ J ) sincefor example, ((0 , ¯1) , (2 , ¯1) / ∈ ¯ Q where Q = h i + Z ∈ M ax ( S ) \ V ( J ) . Since alsoclearly ((1 , ¯0) , (1 , ¯0)) / ∈ I ⋊⋉ J , then I ⋊⋉ J is not a (weakly) J -ideal of R ⋊⋉ J . Similarly, we justify in the following example that if J * J ( R ), then the conversesof (1) and (2) of Corollary 8 are not true in general. Example 3. R = Z (+) Z , K = 0(+) Z and J = h i + Z * J ( R ) . Then K isa (weakly) J -ideal of R . Moreover, if for a, b ∈ R , j, k ∈ J , ( a + j, m ) / ∈ J ( R ) EAKLY J-IDEALS OF COMMUTATIVE RINGS 13 and ( b + k, m ) / ∈ K , then clearly ab + ak + bj + jk = 0 . Take ((2 , ¯0) , (0 , ¯1)) =((2 , ¯0) , (2 , ¯0) + ( − , ¯1)) , ((0 , ¯0) , (1 , ¯0)) ∈ R ⋊⋉ J . Then ((2 , ¯0) , (0 , ¯1))((0 , ¯0) , (1 , ¯0) =((0 , ¯0) , (0 , ¯1)) ∈ ¯ K \ ((0 , ¯0) , (0 , ¯0)) since (0 , ¯1) ∈ K . But, clearly, ((2 , ¯0) , (0 , ¯1)) / ∈ J ( R ⋊⋉ J ) and ((0 , ¯0) , (1 , ¯0) / ∈ ¯ K . Hence, ¯ K is not a (weakly) J-ideal of R ⋊⋉ J . Even if
Ker ( f ) ⊆ J ( R ), the converse of (1) of Theorem 7 need not be true if f ( J ( R )) = J ( S ) + J . Example 4.
Let R = Z (+) Z , S = Z and J = h i the ideal of S . Considerthe homomorphism f : R → S defined by f (( r, m )) = r . Note that J ( S ) = h i , Ker ( f ) = 0(+) Z = J ( R ) and J ( S ) + J = J = f ( J ( R )) . Now, K = h i is a(weakly) J -ideal of S . Moreover, for ( r , m ) , ( r , m ) ∈ R , j, k ∈ J whenever f ( r , m ) + j / ∈ J ( S ) , f ( r , m ) + k / ∈ K , then ( f ( r , m ) + j )( f ( r , m ) + k ) = 0 .Take (( − , , , ((1 , , ∈ R ⋊⋉ f J . Then (( − , , , ,
1) = (( − , , ∈ ¯ K f but (( − , , / ∈ J ( R ⋊⋉ f J ) and ((1 , , / ∈ ¯ K f . Therefore, ¯ K f is not a(weakly) J -ideal of R ⋊⋉ f J . We have proved in section 2 that if I and I are weakly J -ideals of a ring R ,then so is I + J . However, in the next example, we clarify that the product I I need not be a weakly J -ideal. Example 5.
Let R = Z (+) Z and I = J = 0(+) Z . Now, I is a weakly J -ideal of R by Theorem 6 (2) and clearly there are no ( r , m ) , ( r , m ) ∈ R with ( r , m )( r , m ) = (0 , ¯0) , but ( r , m ) / ∈ J ( R ) , ( r , m ) / ∈ I . Since also J = J ( R ) , then I ⋊⋉ J is a weakly J -ideal of R ⋊⋉ J by Corollary 7. On theother hand, ( I ⋊⋉ J ) = I ⋊⋉ J = h (0 , ¯0) i ⋊⋉ J is not a weakly J -ideal. Indeed, ((2 , ¯1) , (2 , ¯0)) , ((0 , ¯2) , (0 , ¯1) ∈ R ⋊⋉ J with ((2 , ¯1) , (2 , ¯0))((0 , ¯2) , (0 , ¯1)) = ((0 , ¯0) , (0 , ¯2)) ∈ I ⋊⋉ J \ ((0 , ¯0) , (0 , ¯0)) but clearly ((2 , ¯1) , (2 , ¯0)) / ∈ J ( R ⋊⋉ J ) and ((0 , ¯2) , (0 , ¯1) / ∈ I ⋊⋉ J . Let R , S , J , f and I be as in Theorem 6 and let T be an ideal of f ( R ) + J . Asa general case of I ⋊⋉ f J , one can verify that if f ( I ) J ⊆ T ⊆ J , then I ⋊⋉ f T := { ( i, f ( i ) + t ) : i ∈ I , t ∈ T } is an ideal of R ⋊⋉ f J . The proof of the following resultis similar to that of (1) of Theorem 6 and left to the reader. Proposition 16.
Let R , S , J , f , I and T as above. If I ⋊⋉ f T is a weakly J -idealof R ⋊⋉ f J , then I is a weakly J -ideal of R . The following example shows that the converse of Proposition 16 is not true ingeneral.
Example 6.
Let R , S , J and I be as in Example 2 and let T = h i (+) Z .Then IJ ⊆ T ⊆ J and I ⋊⋉ T is not a weakly J-ideal of R ⋊⋉ f J . Indeed, ((0 , ¯1) , (4 , ¯1))((1 , ¯0) , (1 , ¯0) = ((0 , ¯1) , (4 , ¯1)) ∈ I ⋊⋉ T \ ((0 , ¯0) , (0 , ¯0)) but clearly ((0 , ¯1) , (4 , ¯1)) / ∈ J ( R ⋊⋉ J ) and ((1 , ¯0) , (1 , ¯0)) / ∈ I ⋊⋉ T . References [1] D. D. Anderson and M. Axtell, When are associates unit multiples, Rocky Mountain Journalof mathematics 34(3) (2004), 811-828.[2] D. D. Anderson, M. Winders, Idealization of a Module, Journal of Commutative Algebra, 1(1) (2009), 3-56.[3] D. D. Anderson, E. Smith, Weakly prime ideals. Houston J. Math, 29 (4) (2003), 831-840.[4] A. Bouvier, Anneaux pr´esimplifiables et anneaux atomiques, C.R. Acad. Sci. Paris S´er.A-B 272 (1971), 992-994. [5] M. D’Anna and M. Fontana, An amalgamated duplication of a ring along an ideal: the basicproperties, J. Algebra Appl. 6 (2007), no. 3, 443–459.[6] M. D’Anna, M. Fontana, The amalgamated duplication of a ring along a multiplicative-canonical ideal, Ark. Mat. 45 (2) (2007), 241-252.[7] M. D’Anna, C.A. Finocchiaro, and M. Fontana, Properties of chains of prime ideals in anamalgamated algebra along an ideal, J. Pure Appl. Algebra 214 (2010), 1633-1641.[8] H. A. Khashan, A. B. Bani-Ata, J -ideals of commutative rings, International ElectronicJournal of Algebra, 29 (2021), 148-164.[9] H. A. Khashan, E. Yetkin Celikel, Quasi J -ideals of commutative rings, (submitted).[10] N. Mahdou, M. Abdou Salam Moutui, Y. Zahir, Weakly prime ideals issued from an amal-gamated algebra, Hacet. J. Math. Stat. 49 (3) (2020), 1159 – 1167.[11] U. Tekir, S. Koc, K. H. Oral, n-Ideals of commutative rings. Filomat, 31(10) (2017), 2933-2941. Department of Mathematics, Faculty of Science, Al al-Bayt University,Al Mafraq,Jordan.
Email address : [email protected]. Department of Electrical-Electronics Engineering, Faculty of Engineering, HasanKalyoncu University, Gaziantep, Turkey.
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