Quasi J-ideals of Commutative Rings
aa r X i v : . [ m a t h . A C ] F e b QUASI J-IDEALS OF COMMUTATIVE RINGS
HANI A. KHASHAN AND ECE YETKIN CELIKEL
Abstract.
Let R be a commutative ring with identity. In this paper, we intro-duce the concept of quasi J -ideal which is a generalization of J -ideal. A properideal of R is called a quasi J -ideal if its radical is a J -ideal. Many character-izations of quasi J -ideals in some special rings are obtained. We characterizerings in which every proper ideal is quasi J -ideal. Further, as a generalizationof presimplifiable rings, we define the notion of quasi presimplifiable rings. Wecall a ring R a quasi presimplifiable ring if whenever a, b ∈ R and a = ab , theneither a is a nilpotent or b is a unit. It is shown that a proper ideal I that iscontained in the Jacobson radical is a quasi J -ideal (resp. J -ideal) if and onlyif R/I is a quasi presimplifiable (resp. presimplifiable) ring. Introduction
Throughout this paper, we shall assume unless otherwise stated, that all ringsare commutative with non-zero identity. We denote the nilradical of a ring R , theJacobson radical of R , the set of unit elements of R , the set of zero-divisors andthe set of all elements that are not quasi-regular in R by N ( R ) , J ( R ) , U ( R ), Z ( R ) , and N Z ( R ) , respectively. In [11], the concept of n -ideals in commutative ringsis defined and studied. A proper ideal I of R is said to be a n -ideal if whenever a, b ∈ R with ab ∈ I and a / ∈ N ( R ), then b ∈ I . Recently, as a generalization of n -ideals, the notion of J -ideals is introduced and investigated in [10]. A properideal I of R is called a J -ideal if whenever a, b ∈ R with ab ∈ I and a / ∈ J ( R ), then b ∈ I .The aim of this article is to extend the notion of J -ideals to quasi J -ideals. Forthe sake of thoroughness, we give some definitions which we will need throughoutthis study. For a proper ideal I a ring R , let √ I = { r ∈ R : there exists n ∈ N with r n ∈ I } denotes the radical of I and ( I : x ) denotes the ideal { r ∈ R : rx ∈ I } .Let M be a unitary R -module. Recall that the idealization R (+) M = { ( r, m ) : r ∈ R, m ∈ M } is a commutative ring with the addition ( r , m ) + ( r , m ) =( r + r , m + m ) and multiplication ( r , m )( r , m ) = ( r r , r m + r m ). Foran ideal I of R and a submodule N of M , it is well-known that I (+) N is anideal of R (+) M if and only if IM ⊆ N [4, Theorem 3.1]. We recall also from [4,Theorem 3.2] that p I (+) N = √ I (+) M , and the Jacobson radical of R (+) M is J ( R (+) M ) = J ( R )(+) M . For the other notations and terminologies that are usedin this article, the reader is referred to [5]. Date : December, 2020.2020
Mathematics Subject Classification.
Key words and phrases. quasi J -ideal, J -ideal, quasi-presimplifiable ring, presimplifiable ring.This paper is in final form and no version of it will be submitted for publication elsewhere. We summarize the content of this article as follows. In Section 2, we study thebasic properties of quasi J -ideals of a ring R . Among many results in this sec-tion, we first start with an example of a quasi J -ideal that is not a J -ideal. InTheorem 1, we give a characterization for quasi J -ideals. In Theorem 2, we con-clude some equivalent conditions that characterize quasi-local rings. The relationsamong primary, δ - n -ideal and quasi J -ideals are clarified (Proposition 2). More-over, Example 2 and Example 3 are presented showing that the converses of theused implications are not true in general. Further, in Theorem 3, we show that ev-ery maximal quasi J -ideal is a J -ideal. In Theorem 4, we characterize quasi J -idealsof zero-dimensional rings in terms of quasi primary ideals. Moreover, the behaviorof quasi J -ideals in polynomial rings, power series rings, localizations, direct prod-uct of rings, idealization rings are investigated (Proposition 13, Proposition 8, andProposition 9, Remark 1 and Proposition 15).In Section 3, we introduce quasi presimplifiable rings as a new generalization ofpresimplifiable rings. We call a ring R quasi presimplifiable if whenever a, b ∈ R with a = ab , then a ∈ N ( R ) or b ∈ U ( R ). Clearly, the classes of presimplifiableand quasi presimplifiable reduced rings coincide. However, in Example 5, we showthat in general this generalization is proper. In Proposition 10, it is shown that aring R is quasi presimplifiable if and only if N Z ( R ) ⊆ J ( R ). The main objectiveof the section is to characterize a J -ideal (resp. a quasi J -ideal) of R as the ideal I for which R/I is a presimplifiable (resp. quasi presimplifiable) ring. This char-acterization is used to justify more results concerning the class of J -ideals (resp.quasi J -ideals). For example, in Theorem 6, it is shown that if { I α : α ∈ Λ } is afamily of J -ideals (resp. quasi J -ideals) over a system of rings { R α : α ∈ Λ } , then I = S α ∈ Λ ϕ α ( I α ) is a J -ideal (resp. quasi J -ideal) of R = lim −→ R α .2. Properties of Quasi J -ideals Definition 1.
Let R be a ring. A proper ideal I of R is said to be a quasi J -idealif √ I is a J -ideal. It is clear that every J -ideal is a quasi J -ideal. However, this generalization isproper and the following is an example of a quasi J -ideal in a certain ring which isnot a J -ideal. Example 1.
Consider the idealization ring R = Z (+) Z . Then I = 0(+) Z is a J -ideal of R since is a J -ideal of Z by [10, Proposition 3.12] . Now, p Z = √ Z = 0(+) Z is a J -ideal of R , and thus Z is a quasi J -ideal of R .However, Z is not a J -ideal of R since for example (0 , , (2 , ∈ R with (2 , · (0 ,
1) = (0 , ∈ Z and (2 , / ∈ J ( R ) = J ( Z )(+) Z = 0(+) Z but (0 , / ∈ Z . Our starting point is the following characterization for quasi J -ideals. Theorem 1.
Let I be a proper ideal of a ring R. Then the following statementsare equivalent: (1) I is a quasi J -ideal of R. (2) If a ∈ R and K is an ideal of R with aK ⊆ I , then a ∈ J ( R ) or K ⊆ √ I. (3) If K and L are ideals of R with KL ⊆ I , then K ⊆ J ( R ) or L ⊆ √ I. (4) If a, b ∈ R and ab ∈ I , then a ∈ J ( R ) or b ∈ √ I. UASI J-IDEALS OF COMMUTATIVE RINGS 3
Proof. (1) ⇒ (2) Suppose that I is a quasi J -ideal of R, aK ⊆ I and a / ∈ J ( R ) . Since √ I is a J -ideal, √ I = ( √ I : a ) by [10, Proposition 2.10]. Thus K ⊆ ( I : a ) ⊆ ( √ I : a ) = √ I. (2) ⇒ (3) Suppose that KL ⊆ I and K * J ( R ) . Then there exists a ∈ K \ J ( R ) . Since aL ⊆ I and a / ∈ J ( R ), we have L ⊆ √ I by our assumption.(3) ⇒ (4) Suppose that a, b ∈ R and ab ∈ I . The result follows by letting K = and L = < b > in (3).(4) ⇒ (1) We show that √ I is a J -ideal. Suppose that ab ∈ √ I and a / ∈ J ( R ).Then there exists a positive integer n such that a n b n ∈ I and a / ∈ J ( R ). It followsclearly that a n / ∈ J ( R ) and so b n ∈ √ I by (4). Therefore, b ∈ p √ I = √ I and I isa quasi J -ideal. (cid:3) As a consequence of Theorem 1, we have the following.
Corollary 1.
Let L be an ideal of a ring R such that L * J ( R ) . Then (1) If I and K are quasi J -ideals of R with IL = KL , then √ I = √ K. (2) If for an ideal I of R , IL is a quasi J -ideal, then √ IL = √ I. 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 quasi J -ideals of R. Proposition 1.
Let I be an ideal of R. Then the following statements are equiva-lent: (1) I is a quasi J -ideal of R. (2) I ⊆ J ( R ) and if whenever a, b ∈ R with ab ∈ I , then a ∈ J ( I ) or b ∈ √ I. Proof. (1) ⇒ (2) Suppose I is a quasi J -ideal of R. Since √ I is a J -ideal, then I ⊆ √ I ⊆ J ( R ) by [10, Proposition 2.2]. Now, (2) follows clearly since J ( R ) ⊆ J ( I ).(2) ⇒ (1) Suppose that ab ∈ I and a / ∈ J ( R ) . Since I ⊆ J ( R ) , we conclude that J ( I ) ⊆ J ( J ( R )) = J ( R ) and so we get a / ∈ J ( I ) . Thus, b ∈ √ I and I is a quasi J -ideal of R. (cid:3) In the following theorem, we characterize rings in which every proper (principal)ideal is a quasi J -ideal. Theorem 2.
For a ring R , the following statements are equivalent: (1) R is a quasi-local ring.(2) Every proper principal ideal of R is a J -ideal.(3) Every proper ideal of R is a J -ideal.(4) Every proper ideal of R is a quasi J -ideal.(5) Every proper principal ideal of R is a quasi J -ideal.(6) Every maximal ideal of R is a quasi J -ideal. Proof. (1) ⇒ (2) ⇒ (3) is clear by [10, Proposition 2.3].Since (3) ⇒ (4) ⇒ (5) is also clear, we only need to prove (5) ⇒ (6) and (6) ⇒ (1).(5) ⇒ (6) Assume that every proper principal ideal of R is a quasi J -ideal. Let M be a maximal ideal of R. Suppose that ab ∈ M and a / ∈ √ M = M. Since < ab > is proper in R , ( ab ) is a quasi J -ideal by our assumption. Since ab ∈ < ab > andclearly a / ∈ √ < ab > , we conclude that b ∈ J ( R ) , as required. HANI A. KHASHAN AND ECE YETKIN CELIKEL (6) ⇒ (1) Let M be a maximal ideal of R . Then M is a quasi J -ideal by (6) whichimplies M = √ M ⊆ J ( R ) by [10, Proposition 2.2]. Thus, J ( R ) = M ; and so R isa quasi-local ring. (cid:3) Let R be a ring and denote the set of all ideals of R by L ( R ). D. Zhao [13]introduced the concept of expansions of ideals of the ring R . A function δ : L ( R ) → L ( R ) is called an ideal expansion if the following conditions are satisfied for anyideals I and J of R :(1) I ⊆ δ ( I ).(2) Whenever I ⊆ J , then δ ( I ) ⊆ δ ( J ).For example, δ : L ( R ) → L ( R ) defined by δ ( I ) = √ I is an ideal expansion ofa ring R . For an ideal expansion δ defined on a ring R , the class of δ - n -ideals hasbeen defined and studied recently in [12]. A proper ideal I of R is called a δ - n -idealif whenever a, b ∈ R and ab ∈ I , then a ∈ N ( R ) or b ∈ δ ( I ). Proposition 2.
Let I be a proper ideal of R . (1) If I is a δ - n -ideal, then I is a quasi J -ideal of R. (2) If I is a primary ideal of R and I ⊆ J ( R ), then I is a quasi J -ideal of R . Proof. (1) Suppose that ab ∈ I and a / ∈ J ( R ). Then a / ∈ N ( R ) as N ( R ) ⊆ J ( R ) . Since I is a δ - n -ideal, we have b ∈ δ ( I ) = √ I . By Theorem 1, we conclude that I is a quasi J -ideal of R. (2) Suppose that ab ∈ I and a / ∈ J ( R ). If b / ∈ √ I , then a ∈ I since I is a primaryideal of R which contradicts the assumption that I ⊆ J ( R ). Therefore, b ∈ √ I and I is a quasi J -ideal by Theorem 1. (cid:3) However, the converses of the implications in Proposition 2 are not true in generalas we can see in the following two examples.
Example 2.
Consider the quasi-local ring Z h i = (cid:8) ab : a, b ∈ Z , ∤ b (cid:9) . Then J ( Z h i ) = h i h i = (cid:8) ab : a ∈ h i , ∤ b (cid:9) is a quasi J -ideal of Z h i by Theorem 2.On the other hand, h i h i is not a δ - n -ideal. Indeed, if we take , ∈ Z h i , then . = ∈ h i h i but / ∈ N ( Z h i ) = 0 Z h i and / ∈ q h i h i = h i h i . Example 3.
Consider the ring C ( R ) of all real valued continuous functions andlet M = { f ∈ C ( R ) : f (0) = 0 } . Then M is a maximal ideal of C ( R ) . Consider thequasi-local ring R = ( C ( R )) M and let I = h x sin x i M . Then I is a quasi J -ideal byTheorem 2. On the other hand I is not primary since for example x sin x ∈ I but x n / ∈ I and sin n x / ∈ I for all integers n . Recall that a ring R is said to be semiprimitive if J ( R ) = 0 . Proposition 3.
Let R be a semiprimitive ring. (1) R is an integral domain if and only if the only quasi J -ideal of R is the zeroideal.(2) If R is not an integral domain, then R has no quasi J -ideals. Proof. (1) Suppose that R is an integral domain. Then it is easy to show that 0is a quasi J -ideal of R. If I is a non-zero quasi J -ideal, then by Proposition 1 wehave I ⊆ J ( R ) = 0 which is a contradiction. UASI J-IDEALS OF COMMUTATIVE RINGS 5 (2) Suppose that I is a quasi J -ideal of R. Then I ⊆ √ I ⊆ J ( R ) = 0. But since R is not integral domain, then 0 is not a prime ideal of R and so clearly it is not aquasi J -ideal. (cid:3) Let R be a ring and S be a non-empty subset of R . Then clearly ( I : S ) = { r ∈ R : rS ⊆ I } is an ideal of R . Now, while it is clear that p ( I : S ) ⊆ (cid:16) √ I : S (cid:17) ,the reverse inclusion need not be true in general. For example, consider S = { } ⊆ Z and the ideal I = h i of Z . Then p ( I : S ) = p h i = h i while (cid:16) √ I : S (cid:17) = h i . Lemma 1. If I is a quasi J -ideal of a ring R and S * J ( R ) is a subset of R , then p ( I : S ) = (cid:16) √ I : S (cid:17) .Proof. If a ∈ (cid:16) √ I : S (cid:17) , then sa ∈ √ I for all s ∈ S . Choose s / ∈ J ( R ) such that sa ∈ √ I . Then a ∈ √ I as I is a quasi J -ideal and so clearly, a ∈ p ( I : S ). Theother inclusion is obvious. (cid:3) Lemma 2.
Let S be a subset of a ring R with S * J ( R ) and I be a proper ideal of R . If I is a quasi J -ideal, then ( I : S ) is a quasi J -ideal.Proof. We first note that ( I : S ) is proper in R since otherwise if 1 ∈ ( I : S ),then S ⊆ I ⊆ J ( R ), a contradiction. Suppose that ab ∈ ( I : S ) and a / ∈ J ( R ) for a, b ∈ R . Then abS ⊆ I and a / ∈ J ( R ) which imply that bS ⊆ √ I by Theorem 1.Thus, b ∈ (cid:16) √ I : S (cid:17) = p ( I : S ) by Lemma 1 and we are done. (cid:3) A quasi J -ideal I of a ring R is called a maximal quasi J -ideal if there is no quasi J -ideal which contains I properly. In the following proposition, we justify that anymaximal quasi J -ideal is a J -ideal. Theorem 3.
Let I be a maximal quasi J -ideal of R . Then I is a J -ideal of R .Proof. Suppose I is a maximal quasi J -ideal of R . Let a, b ∈ R such that ab ∈ I and a / ∈ J ( R ). Then ( I : a ) is a quasi J -ideal of R by Lemma 2. Since I is a maximalquasi J -ideal and I ⊆ ( I : a ), then b ∈ ( I : a ) = I . Therefore, I is a J -ideal of R . (cid:3) If J ( R ) is a quasi J -ideal of a ring R , then clearly it is the unique maximal quasi J -ideal of R . In this case, J ( R ) is a prime ideal of R as can be seen in the followingcorollary. Corollary 2.
Let R be a ring. The following are equivalent: (1) J ( R ) is a J -ideal of R . (2) J ( R ) is a quasi J -ideal of R . (3) J ( R ) is a prime ideal of R . Recall from [9] that a proper ideal of a ring R is called a quasi primary ideal if itsradical is prime. We prove in the following theorem that under a certain conditionon R , quasi primary ideals and quasi J -ideal are the same. Theorem 4.
Let R be a zero-dimensional ring and I be an ideal of R with I ⊆ J ( R ) . Then the following are equivalent: (1) I is a quasi J -ideal of R. HANI A. KHASHAN AND ECE YETKIN CELIKEL (2) I is a quasi primary ideal of R. (3) I = P n for some prime ideal P of R and some positive integer n .(4) ( R, √ I ) is a quasi-local ring. Proof. (1) ⇒ (2) Suppose that ab ∈ √ I and a / ∈ √ I . Then there exists a positivenumber n such that a n b n ∈ I . Since R is zero-dimensional, then every prime idealis maximal and so √ I = J ( I ). Since I is a quasi J -ideal and clearly a n / ∈ J ( I ), weconclude b n ∈ √ I by Theorem 1. Thus b ∈ √ I which shows that √ I is prime asneeded.(2) ⇒ (3) Suppose that I is a quasi primary ideal of R . Then √ I is prime. Since R is zero-dimensional, √ I is a maximal ideal and clearly I = P n for some primeideal P of R and some positive number n .(3) ⇒ (4) Suppose that I = P n for some prime ideal P of R and some positiveinteger n . Then √ I = P is also a maximal ideal. Hence our assumption I ⊆ J ( R )implies that √ I = P = J ( R ) and so ( R, √ I ) is a quasi-local ring.(4) ⇒ (1) It follows directly by Theorem 2. (cid:3) Since every principal ideal ring is zero-dimensional, we have the following corol-lary of Theorem 4.
Corollary 3.
Let R be a principal ideal ring and I be a proper ideal of R . Then I is a quasi J -ideal of R if and only if I = p n R for some prime element p of R with p ∈ J ( R ) and n ≥ . Let I be a proper ideal of R . Then I is said to be superfluous if whenever K isan ideal of R such that I + K = R , then K = R. Proposition 4. If I is a quasi J -ideal of a ring R , then I is superfluous.Proof. Suppose that I + K = R for some ideal K of R . Then √ I + √ K = √ I + K = R. From [10, Proposition 2.9], we conclude that √ K = R which means K = R andwe are done. (cid:3) Proposition 5. (1) If I , I , ..., I k are quasi J -ideals of a ring R , then k \ i =1 I i isa quasi J -ideal of R . (2) Let I , I , ..., I k be quasi primary ideals of a ring R in which their radicalsare not comparable. If k \ i =1 I i is a quasi J -ideal of R , then I i is a quasi J -ideal of R for i = 1 , , ..., k .Proof. (1) Since vuut k \ i =1 I i = k \ i =1 √ I i , the claim is clear by [10, Proposition 2.25].(2) Without loss of generality, we show that I is a quasi J -ideal. Suppose that ab ∈ I and a / ∈ J ( R ) . By assumption, we can choose an element c ∈ k Y i =2 I i ! \√ I and then we have abc ∈ k \ i =1 I i . It follows that bc ∈ vuut k \ i =1 I i = k \ i =1 √ I i ⊆ √ I as UASI J-IDEALS OF COMMUTATIVE RINGS 7 k \ i =1 I i is a quasi J -ideal. Since I is quasi primary, √ I is prime which implies that b ∈ √ I . Thus I is a quasi J -ideal of R. (cid:3) Proposition 6. (1)
Let I , I , ..., I k be quasi J -ideals of a ring R . Then k Q i =1 I i is a quasi J -ideal of R . (2) Let I , I , ..., I k be quasi primary ideals of R in which their radicals are notcomparable. If k Q i =1 I i is a quasi J -ideal of R , then I i is a quasi J -ideal of R for i = 1 , , ..., k. Proof. (1) Let a, b ∈ R such that ab ∈ k Q i =1 I i and a / ∈ J ( R ). Then clearly for all i = 1 , , ..., k , b ∈ √ I i since I i is a quasi J -ideal of R . Now, for all i , there is aninteger n i such that b n i ∈ I i . Thus, b n + n + ··· + n k ∈ k Q i =1 I i and so b ∈ s k Q i =1 I i .Therefore, k Q i =1 I i is a quasi J -ideal.(2) Similar to the proof of Proposition 5 (2). (cid:3) However, the J -ideal property can not pass to the product of ideals as can beseen in the following example. Example 4.
Consider the ring Z (+) Z . Then Z is a J -ideal since is a J -ideal of Z . But (0(+) Z )(0(+) Z ) = 0(+)0 is not a J -ideal of Z (+) Z since forexample, (2 , ,
1) = (0 , and (2 , / ∈ J ( Z )(+) Z = J ( Z (+) Z ) but (0 , =(0 , . Proposition 7.
Let R and R be two rings and f : R → R be an epimorphism.Then the following statements hold: (1) If I is a quasi J -ideal of R with K erf ⊆ I , then f ( I ) is a quasi J -idealof R . (2) If I is a quasi J -ideal of R and K erf ⊆ J ( R ), then f − ( I ) is a quasi J -ideal of R . Proof. (1) Suppose that I is a quasi J -ideal of R . Since √ I is a J -ideal of R and K erf ⊆ I ⊆ √ I , then f ( √ I ) is a J -ideal of R by [10, Proposition 2.23]. Now,if a, b ∈ R such that ab ∈ p f ( I ) and a / ∈ J ( R ), then a n b n ∈ f ( I ) ⊆ f ( √ I )for some integer n . Since a n / ∈ J ( R ), then b n ∈ f ( √ I ) ⊆ p f ( I ). Therefore, b ∈ p f ( I ) and p f ( I ) is a J -ideal of R . So, f ( I ) is a quasi J -ideal of R . (2) Suppose that I is a quasi J -ideal of R . Since √ I is a J -ideal of R and K erf ⊆ J ( R ), then f − ( √ I ) is a J -ideal of R by [10, Proposition 2.23]. Now,let x, y ∈ R such that xy ∈ p f − ( I ) and x / ∈ J ( R ). Then x m y m ∈ f − ( I ) ⊆ f − ( √ I ) for some integer m . But x m / ∈ J ( R ) implies that y m ∈ f − ( √ I ) ⊆ p f − ( I ). It follows that y ∈ p f − ( I ); and so p f − ( I ) is a J -ideal of R . (cid:3) Corollary 4.
Let I and K be proper ideals of R with K ⊆ I . If I is a quasi J -idealof R , then I/K is a quasi J -ideal of R/K . HANI A. KHASHAN AND ECE YETKIN CELIKEL
Proof.
Consider the natural epimorphism π : R → R/K with
Ker ( π ) = K ⊆ I. ByProposition 7, π ( I ) = I/K is a quasi J -ideal of R/K . (cid:3) 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 8.
Let S be a multiplicatively closed subset of a ring R such that J ( S − R ) = S − J ( R ) . Then the following hold: (1) If I is a quasi J -ideal of R such that I ∩ S = ∅ , then S − I is a quasi J -idealof S − R. (2) If S − I is a quasi J -ideal of S − R and S ∩ Z I ( R ) = S ∩ Z J ( R ) ( R ) = ∅ , then I is a quasi J -ideal of R. Proof. (1) Suppose that I is a quasi J -ideal of R. Since √ I is a J -ideal of R , thenby [10, Proposition 2.26], we conclude that √ S − I = S − √ I is a J -ideal of R andwe are done.(2) Let a, b ∈ R and ab ∈ I . Hence a b ∈ S − I . Since S − I is a quasi J -idealof S − R , we have either a ∈ J ( S − ( R )) = S − J ( R ) or b ∈ √ S − I = S − √ I by Theorem 1 . If b ∈ S − √ I , then there exist u ∈ S and a positive integer n such that u n b n ∈ I . Since S ∩ Z I ( R ) = ∅ , we conclude that b n ∈ I and so b ∈ √ I. If a ∈ S − J ( R ), then there exist v ∈ S and a positive integer m suchthat v m a m ∈ J ( R ). Since S ∩ Z J ( R ) ( R ) = ∅ , we conclude that a m ∈ J ( R ) and so a ∈ J ( R ) . Therefore, I is a quasi J -ideal of R by Theorem 1. (cid:3) Next, we justify that decomposable rings have no J -ideals. Remark 1.
Let R and R be two rings and R = R × R . Then there are no quasi J -ideal in R . Indeed, for every proper ideal I × I of R we have (1 , , ∈ I × I but neither (1 , ∈ J ( R ) nor (0 , ∈ √ I × I = √ I × √ I . Lemma 3.
Let I be an ideal of a Noetherian ring R . Then p I [ | x | ] = √ I [ | x | ] .Proof. See [1]. (cid:3)
Proposition 9.
Let I be a proper ideal of a Noetherian ring R . Then I [ | x | ] is aquasi J -ideal of R [ | x | ] if and only if I is a quasi J -ideal of R .Proof. Follows by [10, Proposition 2.18] and Lemma 3. (cid:3) Quasi presimplifiable rings
Recall that a ring R is called presimplifiable if whenever a, b ∈ R with a = ab ,then a = 0 or b ∈ U ( R ). This class of rings has been introduced by Bouvier in[7]. Then many of its properties are studied in [2] and [3]. Among many othercharacterizations, it is well known that R is presimplifiable if and only if Z ( R ) ⊆ J ( R ). As a generalization of presimplifiable property, we introduce the followingclass of rings. Definition 2.
A ring R is called quasi presimplifiable if whenever a, b ∈ R with a = ab , then a ∈ N ( R ) or b ∈ U ( R ) . It is clear that any presimplifiable ring R is quasi presimplifiable and that theycoincide if R is reduced. The following example shows that the converse is not truein general. UASI J-IDEALS OF COMMUTATIVE RINGS 9
Example 5.
Let R = Z (+) Z and let ( a, m ) , ( b, m ) ∈ R such that ( a, m )( b, m ) =( a, m ) and ( a, m ) / ∈ N ( R ) = N ( Z )(+) Z . Then ab = a with a / ∈ N ( R ) and sowe must have b = 1 ∈ U ( Z ) . It follows that ( b, m ) ∈ U ( Z )(+) Z = U ( Z (+) Z ) = U ( R ) and R is quasi presimplifiable. On the other hand, R is not presimplifiable.For example (0 , , (3 , ∈ R and (0 , ,
1) = (0 , but (0 , , (3 , = (0 , and (0 , , (3 , / ∈ U ( R ) . A non-zero element a in a ring R is called quasi-regular if Ann R ( a ) ⊆ N ( R ).We denote the set of all elements of R that are not quasi-regular by N Z ( R ). As acharacterization of quasi presimplifiable rings, we have the following. Proposition 10.
A ring R is quasi presimplifiable if and only if N Z ( R ) ⊆ J ( R ) .Proof. Suppose R is quasi presimplifiable, a ∈ N Z ( R ) and r ∈ R . Then ra ∈ N Z ( R ) and so there exists b / ∈ N ( R ) such that rab = 0. Hence, (1 − ra ) b = b andso by assumption, 1 − ra ∈ U ( R ). It follows that a ∈ J ( R ) and so N Z ( R ) ⊆ J ( R ).Conversely, suppose N Z ( R ) ⊆ J ( R ) and let a, b ∈ R with a = ab . Then a (1 − b ) = 0.If a ∈ N ( R ), then we are done, otherwise, 1 − b ∈ N Z ( R ) ⊆ J ( R ). Therefore, b ∈ U ( R ) as required. (cid:3) The main result of this section is to clarify the relationship between quasi J -ideals(resp. J -ideals) and quasi presimplifiable (resp. presimplifiable) rings. Theorem 5.
Let I be a proper ideal of a ring R . Then (1) I is a J -ideal of R if and only if I ⊆ J ( R ) and R/I is presimplifiable.(2) I is a quasi J -ideal of R if and only if I ⊆ J ( R ) and R/I is quasi presim-plifiable.
Proof. (1) Suppose I is a J -ideal of R . Then I ⊆ J ( R ) by [10, Proposition2.2]. Now, let a + I ∈ Z ( R/I ). Then there exists I = b + I ∈ R/I suchthat ( a + I )( b + I ) = I . Now, ab ∈ I and b / ∈ I imply that a ∈ J ( R )as I is a J -ideal of R . Thus, a + I ∈ J ( R ) /I = J ( R/I ) and so
R/I ispresimplifiable. Conversely, suppose
R/I is presimplifiable and let a, b ∈ R such that ab ∈ I and a / ∈ J ( R ). Then a + I / ∈ J ( R ) /I = J ( R/I ) and byassumption, a + I / ∈ Z ( R/I ). As ( a + I )( b + I ) = I , we conclude that b + I = I and so b ∈ I as needed.(2) Suppose I is a quasi J -ideal of R and note that I ⊆ J ( R ) by Proposition 1.Let a + I ∈ N Z ( R/I ) and choose b + I / ∈ N ( R/I ) such that ( a + I )( b + I ) = I .Then ab ∈ I and b / ∈ √ I which imply that a ∈ J ( R ) as I is a quasi J -idealof R . Hence, a + I ∈ J ( R ) /I = J ( R/I ) and
R/I is quasi presimplifiableby Proposition 10. Conversely, suppose
R/I is quasi presimplifiable and let a, b ∈ R such that ab ∈ I and a / ∈ J ( R ). Then a + I / ∈ J ( R ) /I = J ( R/I )and so a + I / ∈ N Z ( R/I ). As ( a + I )( b + I ) = I , we must have b + I ∈ N ( R/I )and so b ∈ √ I . Therefore, I is a quasi J -ideal. (cid:3) In view of Theorem 5, we deduce immediately the following characterization ofpresimplifiable (resp. quasi presimplifiable) rings.
Corollary 5.
A ring R is presimplifiable (resp. quasi presimplifiable) if and onlyif is a J -ideal (resp. quasi J -ideal) of R . Recall that a ring R is said to be von Neumann regular if for every a ∈ R , thereexists an element x ∈ R such that a = a x . Lemma 4. If R is a quasi presimplifiable von Neumann regular ring, then R is afield.Proof. Let a be a non-zero element of R . Since R is von Neumann regular, a = a x for some element x of R . Observe that a / ∈ N ( R ) as every von Neumann regularring is reduced. Since a = a ( ax ) and R is quasi presimplifiable, we conclude that ax ∈ U ( R ) and so a ∈ U ( R ). Thus, R is a field. (cid:3) We call an ideal I of a ring R regular if R/I is a von Neumann regular ring.
Proposition 11.
Any regular quasi J -ideal in a ring R is maximal.Proof. Suppose I is a regular quasi J -ideal of R . Then R/I is a von Neumannregular ring. Moreover, as I ⊆ J ( R ), then R/I is quasi presimplifiable by Theorem5. It follows by Lemma 4 that
R/I is a field and so I is maximal in R . (cid:3) For a ring R , we recall that f ( x ) = n X i =0 a i x i ∈ R [ x ] is a unit if and only if a ∈ U ( R ) and a , a , ..., a n ∈ N ( R ). In [2], it has been proved that R [ x ] ispresimplifiable if and only if R is presimplifiable and 0 is a primary ideal of R . Proposition 12.
Let R be a ring. Then R [ x ] is quasi presimplifiable if and onlyif R is quasi presimplifiable and is a δ - n -ideal of R .Proof. Suppose that R [ x ] is presimplifiable and let a, b ∈ R ⊆ R [ x ] such that a = ab and a / ∈ N ( R ). Then a / ∈ N ( R [ x ]) and so by our assumption b ∈ U ( R [ x ]).It follows that b ∈ U ( R ) and so R is quasi presimplifiable. Now, let a, b ∈ R suchthat ab = 0 and a / ∈ N ( R ). Then we have a = a (1 − bx ) and so 1 − bx ∈ U ( R [ x ]).Hence b ∈ N ( R ) and 0 is a δ - n -ideal. For the converse, let f ( x ) = n X i =0 a i x i , g ( x ) = m X j =0 b j x j ∈ R [ x ] such that f ( x ) = f ( x ) g ( x ) and f ( x ) / ∈ N ( R [ x ]). Then a i / ∈ N ( R ) for some i . Now, a i = g ( x ) a i implies that a i = b a i and so b ∈ U ( R )since R is quasi presimplifiable. Moreover, for all j = 0, we have a i b j = 0. So, b j ∈ N ( R ) for all j = 0 as 0 is a δ - n -ideal of R . Therefore, g ( x ) ∈ U ( R [ x ]) and R [ x ] is quasi presimplifiable. (cid:3) Recall that a ring R is called a Hilbert ring if every prime ideal of R is anintersection of maximal ideals. Moreover, it is well known that R is a Hilbert ringif and only if M ∩ R is a maximal ideal of R whenever M is a maximal ideal of R [ x ],see [8]. In this case, we have J ( R )[ x ] ⊆ J ( R [ x ]). Indeed, if M is a maximal idealof R [ x ], then M ∩ R is a maximal ideal of R . Hence, J ( R )[ x ] ⊆ ( M ∩ R )[ x ] ⊆ M .In the following proposition, we determine conditions under which the extension I [ x ] in R [ x ] is a quasi J -ideal. Proposition 13.
Let I be an ideal of a ring R . (1) If I [ x ] is a quasi J -ideal of R [ x ], then I is a quasi J -ideal of R .(2) If R is Hilbert, I ⊆ J ( R ) and I is a δ - n -ideal of R , then I [ x ] is a quasi J -ideal of R [ x ]. UASI J-IDEALS OF COMMUTATIVE RINGS 11
Proof. (1) Suppose I [ x ] is a quasi J -ideal of R [ x ] and let a, b ∈ R ⊆ R [ x ]such that ab ∈ I ⊆ I [ x ] and a / ∈ J ( R ). Then clearly a / ∈ J ( R [ x ]) and so b ∈ p I [ x ]. It follows clearly that b ∈ √ I and so I is a quasi J -ideal of R .(2) Suppose I is a δ - n -ideal of R . Then I is a quasi J -ideal by Proposition 2.Thus, R/I is quasi presimplifiable by Theorem 5 (2). By Proposition 12, weconclude that R [ x ] /I [ x ] ∼ = ( R/I )[ x ] is also quasi presimplifiable. Moreover,since R is Hilbert, then I [ x ] ⊆ J ( R )[ x ] ⊆ J ( R [ x ]). Therefore, I [ x ] is a quasi J -ideal of R [ x ] again by Theorem 5 (2). (cid:3) Recall that (Λ , ≤ ) is called a directed quasi-ordered set if ≤ is a reflexive andtransitive relation on Λ and for α, β ∈ Λ, there exists γ ∈ Λ with α ≤ γ and β ≤ γ . A system of rings over (Λ , ≤ ) is a collection { R α : α ∈ Λ } of rings, togetherwith ring homomorphisms ϕ α,β : R α → R β for all α, β ∈ Λ with α ≤ β such that ϕ β,γ ◦ ϕ α,β = ϕ α,γ whenever α ≤ β ≤ γ and such that ϕ α,α = Id R α for all α .A direct limit of { R α : α ∈ Λ } is a ring R together with ring homomorphisms ϕ α : R α → R such that ϕ β ◦ ϕ α,β = ϕ α for all α, β ∈ Λ with α ≤ β and such thatfollowing property is satisfied: For any ring S and collection { f α : α ∈ Λ } of ringmaps f : R α → S such that f β ◦ ϕ α,β = f α for all α, β ∈ Λ with α ≤ β , there is aunique ring homomorphism f : R → S with f ◦ ϕ α = f α for all α ∈ Λ. This directlimit is usually denoted by R = lim −→ R α . Lemma 5. [6]
Let { R α : α ∈ Λ } be a system of rings and let R = lim −→ R α . If { I α : α ∈ Λ } is a family of ideals over { R α : α ∈ Λ } , then I = P α ∈ Λ ϕ α ( I α ) = S α ∈ Λ ϕ α ( I α ) is an ideal of R . Moreover, R/I = lim −→ R α /I α . In [3], it is proved that if { R α : α ∈ Λ } is a system of presimplifiable rings, thenso is R = lim −→ R α . In the following proposition, we generalize this result to quasipresimplifiable case. Proposition 14.
Let (Λ , ≤ ) be a directed quasi-ordered set and let { R α : α ∈ Λ } be a direct system of rings. If each R α is quasi presimplifiable, then the direct limit R = lim −→ R α is quasi presimplifiable.Proof. Let x, y ∈ R with x = xy and x / ∈ N ( R ). For α ∈ Λ, let ϕ α : R α → R be thenatural map. Then there exist α ∈ Λ and x α , y α ∈ R α such that ϕ α ( x α ) = x , ϕ α ( y α ) = y and x α y α = x α . Since x / ∈ N ( R ), then x α / ∈ N ( R α ), see [5], andso y α ∈ U ( R α ) as R α is quasi presimplifiable. Therefore, y = ϕ α ( y α ) ∈ U ( R )and so R is quasi presimplifiable. (cid:3) Theorem 6.
Let (Λ , ≤ ) be a directed quasi-ordered set and let { R α : α ∈ Λ } bea direct system of rings. If { I α : α ∈ Λ } is a family of J -deals (resp. quasi J -ideals) over { R α : α ∈ Λ } , then I = S α ∈ Λ ϕ α ( I α ) is a J -ideal (resp. quasi J -ideal)of R = lim −→ R α .Proof. For all α ∈ Λ, we have I α ⊆ J ( R α ). Hence, I = S α ∈ Λ ϕ α ( I α ) ⊆ S α ∈ Λ ϕ α ( J ( R α )) ⊆ J (lim −→ R α ) = J ( R ). Indeed, let x ∈ S α ∈ Λ ϕ α ( J ( R α )) and r ∈ R . Then there exist α ∈ Λ and x α , r α ∈ R α such that ϕ α ( x α ) = x and ϕ α ( r α ) = r . Now,1 − rx = ϕ α (1 R α − r α x α ) ∈ ϕ α ( U ( R α )) ⊆ U ( R ) and so x ∈ J ( R ). Since for all α ∈ Λ, I α is a J -ideal (quasi J -ideal), then R α /I α is a presimplifiable (quasipresimplifiable) ring by Theorem 5. This implies that R/I = lim −→ R α /I α is presim-plifiable (quasi presimplifiable) by Proposition 14. It follows again by Theorem 5that I is a J -ideal (quasi J -ideal) of R . (cid:3) Finally, for a ring R , an ideal I of R and an R -module M , we determine whenis the ideal I (+) M quasi J -ideal in R (+) M . Proposition 15.
Let I be an ideal of a ring R and let M be an R -module. Then I (+) M is a quasi J -ideal of R (+) M if and only if I is a quasi J -ideal of R .Proof. We have I (+) M ⊆ J ( R (+) M ) if and only if I ⊆ J ( R ) and R/I ∼ = R (+) M/I (+) M .Therefore, the result follows directly by Theorem 5. (cid:3) References [1] M. Achraf, H. Ahmed and B. Ali, 2-absorbing ideals in formal power series rings, PalestineJournal of Mathematics, 6 (2) (2017), 502-506.[2] D. D. Anderson, and S. Valdes-Leon, Factorization in commutative rings with with zerodivisors. Rocky Mountain J. Math, 26 (1996), 439 - 480.[3] D. D. Anderson, M. Axtell, S.J. Forman, J. Stickles, When are associates unit multiples?.Rocky Mountain J. Math. 34 (2004), 811-828.[4] D. D. Anderson, M. Winders, Idealization of a Module, Journal of Commutative Algebra, 1(1) (2019), 3-56.[5] M. F. Atiyah, I. G. MacDonald, Introduction to commutative algebra: Addison-Wesley-Longman, 1969.[6] A. Bell, S. Stalder, M. Teply, Prime ideals and radicals in semigroup-graded rings. Proceedingsof the Edinburgh Mathematical Society, 39 (1) (1996), 1-25.[7] A. Bouvier, Anneaux pr´esimplifiables et anneaux atomiques, C.R. Acad. Sci. Paris S´er.A-B 272 (1971), 992-994.[8] O. Goldman, Hilbert rings and the Hilbert nullstellensatz, Mathematische Zeitschrift, 54(1951), 136–140[9] L. Fuchs, On quasi-primary ideals, Acta Univ. Szeged. Sect. Sci. Math., 11 (1947), 174–183.[10] H. A. Khashan, A. B. Bani-Ata, J -ideals of commutative rings, International ElectronicJournal of Algebra (in press).[11] U. Tekir, S. Koc and K. H. Oral, n -Ideals of commutative rings, Filomat, 31 (10) (2017),2933–2941.[12] E. Yetkin Celikel, G. Ulucak, δ - n -ideals of a commutative ring (submitted).[13] D. Zhao, δ -primary ideals of commutative rings, Kyungpook Math. J.,41 (2001), 17–22. 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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