On 1-absorbing δ-primary ideals
aa r X i v : . [ m a t h . A C ] F e b ON -ABSORBING δ -PRIMARY IDEALS ABDELHAQ EL KHALFI, NAJIB MAHDOU, ¨UNSAL TEK˙IR, AND SUAT KOC¸
Abstract.
Let R be a commutative ring with nonzero identity. Let I ( R ) be the set of all ideals of R and let δ : I ( R ) −→ I ( R ) be a function.Then δ is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J ⊆ I , we have L ⊆ δ ( L ) and δ ( J ) ⊆ δ ( I ). Let δ be an expansion function of ideals of R . In this paper, we introduceand investigate a new class of ideals that is closely related to the classof δ -primary ideals. A proper ideal I of R is said to be a 1-absorbing δ -primary ideal if whenever nonunit elements a, b, c ∈ R and abc ∈ I ,then ab ∈ I or c ∈ δ ( I ) . Moreover, we give some basic properties ofthis class of ideals and we study the 1-absorbing δ -primary ideals ofthe localization of rings, the direct product of rings and the trivial ringextensions. Introduction
Throughout this paper, all rings are assumed to be commutative withnonzero identity and all modules are nonzero unital. If R is a ring, then √ I denotes the radical of an ideal I of R , in the sense of [12, page 17]. Let alsoSpec( R ) denotes the set of all prime ideals of R .The prime ideal, which is an important subject of ideal theory, has beenwidely studied by various authors. Among the many recent generalizationsof the notion of prime ideals in the literature, we find the following, dueto Badawi [2]. A proper ideal I of R is said to be a 2-absorbing ideal ifwhenever a, b, c ∈ R and abc ∈ I , then ab ∈ I or ac ∈ I or bc ∈ I . Inthis case √ I = P is a prime ideal with P ⊆ I or √ I = P ∩ P where P , P are incomparable prime ideals with √ I ⊆ I , cf. [2, Theorem 2.4].Recently, Badawi and Yetkin [4] consider a new class of ideals called theclass of 1-absorbing primary ideals. A proper ideal I of a ring R is calleda 1-absorbing primary ideal of R if whenever nonunit elements a, b, c ∈ R and abc ∈ I , then ab ∈ I or c ∈ √ I . In [13], A. Yassine et. al introducedthe concept of 1-absorbing prime ideals which is a generalization of primeideals. A proper ideal I of R is a 1-absorbing prime ideal if whenever wetake nonunit elements a, b, c ∈ R with abc ∈ I , then ab ∈ I or c ∈ I . In thiscase √ I = P is a prime ideal, cf. [13, Theorem 2.3]. And if R is a ring in Mathematics Subject Classification.
Key words and phrases. δ -primary ideal, δ -primary ideal, trivial ring extension. which exists a 1-absorbing prime ideal that is not prime, then R is a localring, that is a ring with one maximal ideal.Let I ( R ) be the set of all ideals of a ring R . Zhao [14] introduced theconcept of expansion of ideals of R . We recall from [14] that a function δ : I ( R ) −→ I ( R ) is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J ⊆ I , we have L ⊆ δ ( L ) and δ ( J ) ⊆ δ ( I ).Note that there are explanatory examples of expansion functions includedin [14, Example 1.2] and [3, Example 1]. In addition, recall from [14] that aproper ideal I of R is said to be a δ -primary ideal of R if whenever a, b ∈ R with ab ∈ I , we have a ∈ I or b ∈ δ ( I ), where δ is an expansion functionof ideals of R . Also, recall from [5] that a proper ideal I of R is called a δ -semiprimary ideal of R if ab ∈ I implies a ∈ δ ( I ) or b ∈ δ ( I ). In thispaper, we introduce and investigate a new concept of ideals that is closelyrelated to the class of δ -primary ideals. A proper ideal I of R is said tobe a 1-absorbing δ -primary ideal if whenever nonunit elements a, b, c ∈ R and abc ∈ I , then ab ∈ I or c ∈ δ ( I ) . For example, let δ : I ( R ) −→ I ( R )such that δ ( I ) = √ I for each ideal I of R . Then δ is an expansion functionof ideals of R , and hence a proper ideal I of R is a 1-absorbing δ -primaryideal of R if and only if I is a 1-absorbing primary ideal of R . Among manyresults in this paper are given to disclose the relations between this newclass and others that already exist. The reader may find it helpful to keepin mind the implications noted in the following figure.1-absorbing prime ideal 1-absorbing δ -primary idealprime ideal ✲ ✲ Among other things, we give an example of 1-absorbing δ -primary idealthat is not 1-absorbing prime ideal (Example 2.3). Also, we show (Theorem2.6) that if a ring R admits a 1-absorbing δ -primary ideal of R that is nota δ -primary ideal, then R is a local ring. Moreover, we prove that if R isa chained ring with maximal ideal M , then the only 1-absorbing δ -primaryideals of R are M and the δ -primary ideals of R (Theorem 2.10). Finally,we give an idea about some 1-absorbing δ -primary ideal of the localizationof rings, the direct product of rings and the trivial ring extensions.2. Main Results
We start this section by the following definition.
Definition 2.1.
A proper ideal I of a ring R is called a -absorbing δ -primary ideal if whenever abc ∈ I for some nonunit elements a, b, c ∈ I ,then ab ∈ I or c ∈ δ ( I ) . Remark 2.2.
Let R be a ring, I a proper ideal of R and δ be an expansionfunction of I ( R ) . (1) If δ ( I ) = I , then I is a -absorbing δ -primary ideal of R if and onlyif it is a -absorbing prime ideal. N 1-ABSORBING δ -PRIMARY IDEALS 3 (2) If δ ( I ) = √ I , then I is a -absorbing δ -primary ideal of R if andonly if it is a -absorbing primary ideal. (3) Every -absorbing prime ideal is a -absorbing δ -primary ideal. (4) Every δ -primary ideal is a -absorbing δ -primary ideal. (5) Let γ be an expansion function of the ideal of R such that δ ( I ) ⊆ γ ( I ) . If I is a -absorbing δ -primary ideal of R , then I is a -absorbing γ -primary ideal of R . Next, we give an example of a 1-absorbing δ -primary ideal that is not a1-absorbing prime ideal. Example 2.3.
Let R := K [[ X , X , X ]] be a ring of formal power serieswhere K is a field. Consider the expansion function δ : I ( R ) −→ I ( R ) defined by δ ( I ) = I + M where M = ( X , X , X ) is the maximal ideal of R . Let I = ( X X X ) be an ideal of R . Thus, I is not a -absorbing primeideal of R since X X X ∈ I but neither X X ∈ I nor X ∈ I . Now, let x, y, z be nonunit elements of R such that xyz ∈ I . Clearly I is a -absorbing δ -primary because z ∈ δ ( I ) = M . Proposition 2.4. (i) Every 1-absorbing δ -primary ideals is also a 2-absorbing δ -primary ideal of R .(ii) Let I be a 1-absorbing δ -primary ideal of R and δ ( I ) be a radical ideal,that is, p δ ( I ) = δ ( I ) . Then I is a δ -semiprimary ideal of R .Proof. (i) Let abc ∈ I for some a, b, c ∈ R . If at least one of a, b, c is aunit of R , then we are done. So assume that a, b, c are nonunits R . Since I is a 1-absorbing δ -primary ideal of R , we get ab ∈ I or c ∈ δ ( I ), whichimplies that ab ∈ I or ac ∈ δ ( I ) or bc ∈ δ ( I ). Therefore, I is a 2-absorbing δ -primary ideal of R .(ii) Suppose that ab ∈ I for some a, b ∈ R . Then we may assume that a, b are nonunits. Thus a b ∈ I implies that a ∈ I or b ∈ δ ( I ). Then we have a ∈ √ I ⊆ p δ ( I ) = δ ( I ) or b ∈ δ ( I ). Hence, I is a δ -semiprimary ideal of R . (cid:3) The converse of previous theorem (i) is not true in general. See thefollowing example.
Example 2.5. (2-absorbing δ -primary ideal that is not 1-absorbing δ -primary ideal) Let R = Z , I = pq Z , where p = q are prime numbers, and δ ( I ) = I + p Z .Since I is a 2-absorbing ideal, so is 2-absorbing δ -primary. However, it iseasy to see that ppq ∈ I , p / ∈ I and q / ∈ δ ( I ) . Thus, I is not a 1-absorbing δ -primary ideal of R . In the next result, we show that if a ring R admits a 1-absorbing δ -primaryideal that is not a δ -primary ideal, then R is a local ring. Theorem 2.6.
Let δ be an ideal expansion. Suppose that a ring R admits a -absorbing δ -primary ideal that is not a δ -primary ideal. Then R is a localring. ABDELHAQ EL KHALFI, NAJIB MAHDOU, ¨UNSAL TEK˙IR, AND SUAT KOC¸
Proof.
Assume that I is a 1-absorbing δ -primary ideal that is not a δ -primaryideal of R. Hence there exist nonunit elements a, b ∈ R such that ab ∈ I , a / ∈ I and b / ∈ δ ( I ). Let d be a nonunit element of R. As dab ∈ I , I is a1-absorbing δ -primary ideal of R and b / ∈ δ ( I ) , we conclude that da ∈ I. Let c be a unit element of R. Suppose that d + c is a nonunit element of R. Since( d + c ) ab ∈ I , I is a 1-absorbing δ -primary ideal of R and b / ∈ δ ( I ), we getthat ( d + c ) a = da + ca ∈ I. Since da ∈ I, we conclude that a ∈ I, whichgives a contradiction. Hence, d + c is a unit element of R . Now, the resultfollows from [4, Lemma 1]. (cid:3) Next, we give a method to construct 1-absorbing δ -primary ideals thatare not δ -primary ideals. Theorem 2.7.
Let R be a local ring with maximal ideal M and δ be an idealexpansion. Let x be a nonzero prime element of R such that δ ( xM ) ( M .If x ∈ δ ( xM ) , then xM is a -absorbing δ -primary ideal of R that is not a δ -primary ideal of R .Proof. First, we will show that xM is a 1-absorbing δ -primary ideal of R .Assume that abc ∈ xM for some nonunit elements a, b, c ∈ R . If ab / ∈ xM ,then a / ∈ xR and b / ∈ xR , so ab / ∈ xR because x is a prime element of R .Moreover, the fact that abc ∈ xR and ab / ∈ xR implies that c ∈ xR ⊆ δ ( xM ).Now, we prove that xM is not a δ -primary ideal of R . By hypothesis, wecan pick an element a ∈ M \ δ ( xM ), hence xa ∈ xM . However, x / ∈ xM since x is an irreducible element of R by [4, Lemma 2]. Which implies that xM is not a δ -primary ideal, this completes the proof. (cid:3) Theorem 2.8.
Let I be a -absorbing δ -primary ideal of a ring R where δ is an ideal expansion and let d ∈ R \ I be a nonunit element of R . Then ( I : d ) = { x ∈ R | dx ∈ I } is a δ -primary ideal of R .Proof. Suppose that ab ∈ ( I : d ) for some elements a, b ∈ R . Without loss ofgenerality, we may assume that a and b are nonunit elements of R . Supposethat a / ∈ ( I : d ). Since dab ∈ I and I is a 1-absorbing δ -primary ideal of R , we conclude that b ∈ δ ( I ). So, b ∈ δ (( I : d )) and this completes theproof. (cid:3) Proposition 2.9.
Let R be a ring, δ an ideal expansion and I be a properideal of R . If I is a -absorbing δ -primary ideal of R , then either I is a δ -semiprimary ideal of R or R is local, say with maximal ideal M , such that M ⊆ I .Proof. If R is not local, then Theorem 2.6 implies that I is δ -primary and so I is a δ -semiprimary ideal of R . Now, assume that R is local with maximalideal M such that I is not a δ -semiprimary ideal of R . Since I is proper,we infer that I ⊆ M . Moreover, there are a, b ∈ M \ δ ( I ) such that ab ∈ I .To prove that M ⊆ I , it suffices to show that xy ∈ I for all x, y ∈ M .Let x, y ∈ M . Then xyab ∈ I . Since xy, a, b ∈ M , b δ ( I ) and I isa 1-absorbing δ -primary ideal, we conclude that xya ∈ I . Again, since N 1-ABSORBING δ -PRIMARY IDEALS 5 x, y, a ∈ M , a δ ( I ) and I is a 1-absorbing δ -primary ideal, we have that xy ∈ I . (cid:3) Recall that a ring R is a chained ring if the set of all ideals of R is linearlyordered by inclusion. Moreover, R is said to be an arithmetical ring if R M is a chained ring for each maximal ideal M of R . We next determinate the1-absorbing δ -primary ideals of a chained ring. Theorem 2.10.
Let R be a chained ring with maximal ideal M , δ an idealexpansion and I be a proper ideal of R such that I = M . Then I is a -absorbing δ -primary ideal of R if and only if I is a δ -primary ideal of R .Proof. We need only prove the “only if” assertion. Let I be a 1-absorbing δ -primary ideal. Thus, Proposition 2.9 gives that either I is a δ -semiprimaryideal of R or M ⊆ I . First, assume that I is a δ -semiprimary ideal of R and ab ∈ I for some nonunit elements a, b ∈ R such that b / ∈ δ ( I ). Hence, a ∈ δ ( I ). Now, since R is a chained ring, we conclude that a ∈ bR and thus a = br for some nonunit element r ∈ R . As brb ∈ I , b / ∈ δ ( I ) and I is a1-absorbing δ -primary ideal of R , we conclude that a = br ∈ I . Which givesthat I is a δ -primary ideal of R . Now, we suppose that M ⊆ I . We mayassume that M = I . Thus, we can pick a ∈ M \ I and b ∈ I \ M . Then b ∈ aR since R is a chained ring. So, b = ar for some nonunit element r ∈ R and thus b ∈ M , a contradiction. This completes the proof. (cid:3) In view of Theorem 2.10, we have the following result.
Corollary 2.11.
Let R be an arithmetical ring with Jacobson radical M and I be a proper ideal of R such that I = M . Then I is a -absorbing δ -primary ideal of R if and only if I is a δ -primary ideal of R .Proof. Assume that R is local with maximal ideal M . Since R is an arith-metical ring, we conclude that R = R M is a chained ring and thus the claimfollows from Theorem 2.10. In the remaining case, suppose that R is not alocal ring. Then the result follows by Theorem 2.6. (cid:3) Proposition 2.12.
Let R be a local ring with principal maximal ideal M , δ an ideal expansion and I be a proper ideal of R . Then I is a -absorbing δ -primary ideal of R if and only if either I is a δ -primary ideal of R or M ⊆ I .In addition, if √ I ⊆ δ ( I ) then I is a -absorbing δ -primary ideal of R if andonly if I is a δ -primary ideal of R .Proof. By Remark 2.2(4) and Proposition 2.9, we need only prove that if I is a 1-absorbing δ -primary ideal of R which is a δ -semiprimary ideal then I is a δ -primary ideal (along with the hypothesis that R be a local ringwith principal maximal ideal M ). Also, we may assume that δ ( I ) = R . Set M = xR and let a and b be nonunit elements of R such that b / ∈ δ ( I ) and ab ∈ I . Since I is a δ -semiprimary ideal of R , we get that a ∈ δ ( I ). Moreover, a = rx for some r ∈ R . If r is a unit element of R then M = δ ( I ) and thus I ABDELHAQ EL KHALFI, NAJIB MAHDOU, ¨UNSAL TEK˙IR, AND SUAT KOC¸ is a δ -primary ideal. If r is a nonunit element of R then rxb = ab ∈ I . Thatimplies a = rx ∈ I since I is a 1-absorbing δ -primary ideal. This completesthe proof. The in addition statement is clear. (cid:3) Proposition 2.13.
Let { J i | i ∈ D } be a directed set of -absorbing δ -primary ideals of R , where δ is an ideal expansion. Then the ideal J = ∪ i ∈ D J i is a -absorbing δ -primary ideal of R .Proof. Let abc ∈ J , then abc ∈ J i for some i ∈ D . Since J i is a 1-absorbing δ -primary ideal of R , ab ∈ J i or c ∈ δ ( J i ) ⊆ δ ( J ). Hence, J is a 1-absorbing δ -primary ideal of R . (cid:3) Proposition 2.14.
Let I be a -absorbing δ -primary ideal of R such that p δ ( I ) = δ ( √ I ) , where δ is an ideal expansion. Then, √ I is a δ -primaryideal of R .Proof. Let ab ∈ √ I such that a / ∈ √ I . Hence, there exists a positive integer n such that ( ab ) n ∈ I . So, a m a m b n ∈ I for some positive integer m . Since I is a 1-absorbing δ -primary ideal of R and a m / ∈ I , we conclude that b n ∈ δ ( I ). That implies b ∈ p δ ( I ) = δ ( √ I ) and so √ I is a δ -primary idealof R . (cid:3) Proposition 2.15.
Let I be a proper ideal of a ring R and δ be an idealexpansion such that δ ( δ ( I )) = δ ( I ) . Then δ ( I ) is a -absorbing δ -primaryideal of R if and only if δ ( I ) is a -absorbing prime ideal of R Proof.
By Remark 2.2(3), we need only prove the “only if” assertion. Let abc ∈ δ ( I ) for some nonunit elements a, b, c ∈ R . Hence ab ∈ δ ( I ) or c ∈ δ ( δ ( I )) = δ ( I ). Thus δ ( I ) is a 1-absorbing prime ideal of R . (cid:3) Proposition 2.16.
Let R be a ring, I a proper ideal of R and δ be an idealexpansion. Then I is a -absorbing δ -primary ideal if and only if whenever I I I ⊆ I for some proper ideals I , I and I of R , then I I ⊆ I or I ⊆ δ ( I ) .Proof. It suffices to prove the “if” assertion. Suppose that I is a 1-absorbing δ -primary ideal and let I , I and I be proper ideals of R such that I I I ⊆ I and I δ ( I ). Thus abc ∈ I for every a ∈ I , b ∈ I and c ∈ I \ δ ( I ). Since I is a 1-absorbing δ -primary ideal, we then have I I ⊆ I , as desired. (cid:3) Recall from [14] that an ideal expansion δ is said to be intersection pre-serving if δ ( I ∩ I ∩ ... ∩ I n ) = δ ( I ) ∩ δ ( I ) ∩ ... ∩ δ ( I n ) for any ideals I , ..., I n of R . Proposition 2.17.
Let δ be an intersection preserving ideal expansion. If I , I , ..., I n are -absorbing δ -primary ideals of R , and δ ( I i ) = P for all i ∈ { , , ..., n } , then I ∩ I ∩ ... ∩ I n is a -absorbing δ -primary ideal of R .Proof. Let abc ∈ J = I ∩ I ∩ ... ∩ I n such that ab / ∈ J . Let i ∈ { , , ..., n } such that ab / ∈ I i . Since abc ∈ I i and I i is a 1-absorbing δ -primary ideal,we conclude that c ∈ δ ( I i ) = δ ( J ). Therefore, J is a 1-absorbing δ -primaryideal of R . (cid:3) N 1-ABSORBING δ -PRIMARY IDEALS 7 Proposition 2.18.
Let R be a ring and δ be an expansion function of I ( R ) .Then the following statements are equivalent: (1) Every proper principal ideal is a -absorbing δ -primary ideal of R . (2) Every proper ideal is a -absorbing δ -primary ideal of R .Proof. Assume that (1) holds and let I be a proper ideal of R . Let a, b, c benonunit elements of R such that abc ∈ I . Hence abc ∈ abcR which impliesthat ab ∈ abcR ⊆ I or c ∈ abcR ⊆ δ ( I ). Therefore I is a 1-absorbing δ -primary ideal of R . The converse is clear. (cid:3) An expansion function δ of I ( R ) is said to satisfy condition ( ∗ ) if δ ( I ) = R for each proper ideal I of R . Note that the identity function and radicaloperation are examples of expansion functions satisfying condition ( ∗ ). Theorem 2.19.
Let R be a ring and δ an expansion function of I ( R ) sat-isfying condition ( ∗ ) and δ ( J ac ( R )) = J ac ( R ) . Suppose that δ ( xI ) = xδ ( I ) for every proper ideal I of R and every x ∈ R . The following statements areequivalent.(i) Every proper principal ideal is a 1-absorbing δ -primary ideal of R .(ii) Every proper ideal is a 1-absorbing δ -primary ideal of R .(iii) R is local with J ac ( R ) = (0) , where J ac ( R ) = m is the uniquemaximal ideal.Proof. ( i ) ⇔ ( ii ) Follows from Proposition 2.18.( i ) ⇒ ( iii ) Assume that every proper ideal is a 1-absorbing δ -primaryideal R . Choose x, y ∈ J ac ( R ). Now, we will show that xy = 0. If x or y is zero, then we are done. Assume that x, y = 0. Since x y ∈ ( x y )and ( x y ) is a 1-absorbing δ -primary ideal, we conclude that x ∈ ( x y )or y ∈ δ (( x y )) = yδ (( x )). Suppose that y ∈ yδ (( x )). Then there exists a ∈ δ (( x )) ⊆ δ ( J ac ( R )) = J ac ( R ) such that y = ya . Which implies that y (1 − a ) = 0. Since 1 − a is unit, we have y = 0, which is a contradiction. Thuswe have, x ∈ ( x y ). Then we can write x = rx y for some r ∈ R . Thisimplies that x (1 − ry ) = 0. Since 1 − ry is unit, we have x = 0. Likewise,we get y = 0. Now, choose another z ∈ J ac ( R ). Since xyz ∈ ( xyz ) and( xyz ) is a 1-absorbing δ -primary, we get xy ∈ ( xyz ) or z ∈ δ (( xyz )) = zδ (( xy )). First, assume that xy ∈ ( xyz ). Then there exists r ∈ R such that xy = rxyz , which implies that xy (1 − rz ) = 0. Since 1 − rz is unit, we have xy = 0 which completes the proof. Now, assume that xy / ∈ ( xyz ), that is, z ∈ δ (( xyz )) = zδ (( xy )). Then there exists a ∈ δ (( xy )) ⊆ J ac ( R ) such that z = za . This implies that z (1 − a ) = 0 so that z = 0. Now, choose z = x + y .Then by above argument, we have either xy = 0 or z = x + y = 0. If z = x + y = 0, then we have x = − y and so xy = − y = 0 which completesthe proof. Therefore, J ac ( R ) = (0).Now, we will show that R is a local ring. Choose maximal ideals M , M of R . Now, put I = M ∩ M . Since M M ⊆ I and I is a 1-absorbing δ -primary ideal, we have either M ⊆ I ⊆ M or M ⊆ δ ( I ) ⊆ δ ( M ). Case1:
Suppose that M ⊆ M . Since M is prime, clearly we have M ⊆ M which implies that M = M . Case 2:
Suppose that M ⊆ δ ( M ). Since δ satisfies condition ( ∗ ), δ ( M ) is proper. As M ⊆ δ ( M ) and M is amaximal ideal, we have M = δ ( M ). Then we get M ⊆ M , which impliesthat M = M . Therefore, R is a local ring.( iii ) ⇒ ( i ) Suppose that R is a local ring with J ac ( R ) = (0). Let I be a proper ideal of R and abc ∈ I for some nonunits a, b, c ∈ R . Then a, b, c ∈ J ac ( R ) since R is local. As J ac ( R ) = (0), we have ab = 0 ∈ I .Therefore, I is a 1-absorbing δ -primary ideal of R . (cid:3) It can be easily seen that, in Theorem 2.19, ( iii ) always implies ( i ) withoutany assumption on δ . But we give some examples showing that the converseis not true if we drop the aforementioned assumptions on δ . Example 2.20.
Let R = Z p , where p is a prime number and δ ( I ) = R forevery proper ideal I of R . Note that δ does not satisfy condition ( ∗ ) andnote that every ideal I of R is 1-absorbing δ -primary. Thus J ac ( R ) = (0) ,while R is a local ring. Example 2.21.
Let k be a field and consider the formal power series ring R = k [[ X ]] . Then R is a local ring with unique maximal ideal m = ( X ) .Define expansion function δ as δ ( I ) = √ I for every ideal I of R . Then it iseasy to see that every ideal of R is a 1-absorbing δ -primary ideal. Also, it isclear that δ satisfies condition ( ∗ ) and δ ( J ac ( R )) = J ac ( R ) but not satisfythe condition δ ( xI ) = xδ ( I ) . Furthermore, J ac ( R ) = (0) . Thus Theorem2.19 fails without assumption δ ( xI ) = xδ ( I ) . Corollary 2.22.
Let R be a ring. The following statements are equivalent.(i) Every proper ideal is a 1-absorbing prime ideal of R .(ii) Every proper principal ideal is a 1-absorbing prime ideal of R .(iii) R is local with J ac ( R ) = (0) .Proof. ( i ) ⇔ ( ii ) Follows from Proposition 2.18.( ii ) ⇒ ( iii ) Let δ be the identity expansion function, that is, δ ( I ) = I forevery ideal I of R . Note that δ satisfies all axioms in Theorem 2.19. Then R is a local ring with J ac ( R ) = (0).( iii ) ⇒ ( i ) It is similar to Theorem 2.19 ( iii ) ⇒ ( i ). (cid:3) An ideal expansion δ is called a prime expansion if for any 1-absorbing δ -primary ideal I of R , δ ( I ) is a prime ideal of R . Proposition 2.23.
Let R be a local ring with maximal ideal M and δ be aprime expansion function of I ( R ) . Assume that one of the following condi-tions holds: (1) Spec ( R ) = { δ (0) } . (2) Spec ( R ) = { δ (0) , M } and δ (0) M = 0 .Then every proper ideal of R is -absorbing δ -primary.Proof. Let I be a proper ideal of R and assume that (1) holds. Since δ is a prime expansion, we conclude that δ ( I ) = δ (0) is the maximal ideal N 1-ABSORBING δ -PRIMARY IDEALS 9 of R . Clearly I is a 1-absorbing δ -primary ideal of R . Now, assume thatSpec( R ) = { δ (0) , M } and δ (0) M = 0. If δ ( I ) = M , we have then I is1-absorbing δ -primary. In the remaining case, δ ( I ) = δ (0). Let abc ∈ I forsome nonunit elements a, b, c ∈ R such that c / ∈ δ (0). As I ⊆ δ (0), we getthat either a ∈ δ (0) or b ∈ δ (0). Thus ab = 0 ∈ I which gives that I is a1-absorbing δ -primary ideal of R . (cid:3) Let f : R → S be a ring homomorphism and δ , γ expansion func-tions of I ( R ) and I ( S ) respectively. Recall from [3] that f is called a δγ -homomorphism if δ ( f − ( I )) = f − ( γ ( I )) for each ideal I of S . Also notethat if f is a δγ -epimorphism and I is an ideal of R containing ker ( f ), then γ ( f ( I )) = f ( δ ( I )). Theorem 2.24.
Let f : R → S be a ring δγ -homomorphism where δ , γ are expansion functions of I ( R ) and I ( S ) respectively. Suppose that f( a ) isnonunit in S for every nonunit element a in R . Then the following state-ments hold. (1) If J is a -absorbing γ -primary ideal of S, then f − ( J ) is a -absorbing δ -primary ideal of R . (2) If f is an epimorphism and I is a proper ideal of R containing ker( f ) , then I is a -absorbing δ -primary ideal of R if and only if f ( I ) is a -absorbing γ -primary ideal of S .Proof. (1) Assume that abc ∈ f − ( J ) , for some nonunit elements a, b, c ∈ R. Then f ( a ) f ( b ) f ( c ) ∈ J . Thus f ( a ) f ( b ) ∈ J or f ( c ) ∈ γ ( J ) , which impliesthat ab ∈ f − ( J ) or c ∈ f − ( γ ( J )) = δ ( f − ( J )) . Therefore, f − ( J ) is a1-absorbing δ -primary ideal of R .(2) Suppose that f ( I ) is an 1-absorbing γ -primary ideal of S . Since I = f − ( f ( I )), we conclude that I is a 1-absorbing δ -primary ideal of R by (1).Conversely, let x, y, z be nonunit elements of S with xyz ∈ f ( I ). Thenthere exist a, b, c ∈ R such that x = f ( a ), y = f ( b ) and z = f ( c ) with f ( abc ) = xyz ∈ f ( I ) . Since ker( f ) ⊆ I , we then have abc ∈ I. Since I is a1-absorbing δ -primary ideal of R and abc ∈ I , we conclude that ab ∈ I or c ∈ δ ( I ) which gives that xy ∈ f ( I ) or z ∈ f ( δ ( I )) = γ ( f ( I )). Thus f ( I ) isa 1-absorbing δ -primary ideal of S . (cid:3) Let δ be an expansion function of I ( R ) and I an ideal of R . Then thefunction ¯ δ : RI −→ RI defined by ¯ δ ( JI ) = δ ( J ) I for all ideals I ⊆ J , becomesan expansion function of RI . Then, we have the following result. Corollary 2.25.
Let R be a ring, δ an expansion function of I ( R ) and I ⊆ J be proper ideals of R . Assume that a + I is a nonunit element of RI for every nonunit element a ∈ R . Then J is a -absorbing δ -primary idealof R if and only if JI is a -absorbing ¯ δ -primary ideal of RI . Proposition 2.26.
Let S be a multiplicatively closed subset of a ring R and δ S an expansion function of I ( S − R ) such that δ S ( S − I ) = S − ( δ ( I )) for each ideal I of R . If I is a 1-absorbing δ -primary ideal of R such that I ∩ S = ∅ , then S − I is a 1-absorbing δ S -primary ideal of S − R .Proof. Let I be a 1-absorbing δ -primary ideal of R such that I ∩ S = ∅ and as bt cr ∈ S − I for some nonunit elements a, b, c ∈ R and s, t, r ∈ S suchthat as bt / ∈ S − I . Then xabc ∈ I for some x ∈ S . Since I is a 1-absorbing δ -primary and xab / ∈ I , we conclude that c ∈ δ ( I ). Thus cr ∈ S − ( δ ( I )) = δ S ( S − I ) which completes the proof. (cid:3) Let S be a multiplicatively closed subset of a ring R and I an ideal of R .The next example shows that if S − I is a 1-absorbing δ S -primary ideal of S − R , then I need not to be a 1-absorbing δ -primary ideal of R . Example 2.27.
Let p = q be two prime numbers. Set I = pq Z and δ bean ideal expansion such that δ ( I ) = I + q Z for each ideal I of Z . Clearly, I is not a 1-absorbing δ -primary ideal of Z because qqp ∈ I but neither q ∈ I nor p ∈ δ ( I ) . Now, let S = Z \ p Z and note that S − I = S − ( p Z ) .Let ar br cr ∈ S − I for some nonunit elements ar , br , cr ∈ S − Z . Note that xr ∈ S − Z is nonunit if and only if x ∈ p Z . Thus a ∈ p Z and b ∈ p Z . Whichgives that ar br ∈ S − I and hence S − I is a -absorbing δ S -primary ideal. Let R and R be two rings, let δ i be an expansion function of I ( R i ) foreach i ∈ { , } and R = R × R . For a proper ideal I × I , the function δ × defined by δ × ( I × I ) = δ ( I ) × δ ( I ) is an expansion function of I ( R ) . The following result characterizes the 1-absorbing δ -primary ideals of thedirect product of rings. Theorem 2.28.
Let R and R be rings, R = R × R and let δ i be anexpansion function of I ( R i ) for i = 1 , . Then the following statements areequivalent: (1) I is a -absorbing δ × -primary ideal of R . (2) I is a δ × -primary ideal of R . (3) Either I = I × R , where I is a δ -primary ideal of R or I = R × I , where I is a δ -primary ideal of R or I = I × I , where I and I are proper ideals of R , R , respectively with δ ( I ) = R and δ ( I ) = R .Proof. (1) ⇔ (2). This follows from Theorem 2.6.(2) ⇔ (3) Let I be a δ × -primary ideal of R. Hence I has the form I = I × I where I and I are ideals of R and R respectively. Without lossof generality, we may assume that I = I × R for some proper ideal I of R . We show that I is a δ -primary ideal of R . Deny. Then there are a, b ∈ R such that ab ∈ I , a / ∈ I and b / ∈ δ ( I ) . Hence ( a, b, ∈ I × R . Which implies that ( a, ∈ I × R or ( b, ∈ δ × ( I × R ) and so a ∈ I or b ∈ δ ( I ) , which gives a contradiction. Now suppose that both I and I are proper. As (1 , , ∈ I × I and (1 , , (0 , / ∈ I × I , we have(1 , , (0 , ∈ δ × ( I × I ) = δ ( I ) × δ ( I ) . Therefore δ ( I ) = R and δ ( I ) = R . The converse is clear. (cid:3) N 1-ABSORBING δ -PRIMARY IDEALS 11 The following example proves that the condition “ δ ( I i ) = P for all i ∈{ , , ..., n } ” is necessary in Proposition 2.17. Example 2.29.
Consider R = Z × Z , I = 4 Z × Z and I = Z × Z . Let δ be an expansion function of I ( Z ) such that for every ideal I of Z we have δ ( I ) = √ I + J where J = 2 Z . Thus δ × ( I ) = 2 Z × Z and δ × ( I ) = Z × Z .Moreover, I and I are -absorbing δ × -primary ideal. But I ∩ I = 4 Z × Z is not a -absorbing δ × -primary ideal by Theorem 2.28. Let A be a ring and E an A -module. Then A ⋉ E , the trivial ( ring ) extension of A by E , is the ring whose additive structure is that of the ex-ternal direct sum A ⊕ E and whose multiplication is defined by ( a, e )( b, f ) :=( ab, af + be ) for all a, b ∈ A and all e, f ∈ E . (This construction is also knownby other terminology and other notation, such as the idealization A (+) E .)The basic properties of trivial ring extensions are summarized in the books[10], [9]. Trivial ring extensions have been studied or generalized extensively,often because of their usefulness in constructing new classes of examples ofrings satisfying various properties (cf. [1, 6, 7, 11] ). In addition, for anideal I of A and a submodule F of E , I ⋉ F is an ideal of A ⋉ E if and onlyif IE ⊆ F . Moreover, for an expansion function δ of A , it is clear that δ ⋉ defined as δ ⋉ ( I ⋉ F ) = δ ( I ) ⋉ E is an expansion function of A ⋉ E . Also asusual, if c ∈ A then ( F : c ) = { e ∈ E | ce ∈ F } . Theorem 2.30.
Let A be a ring, E an A -module and δ be an expansionfunction of I ( A ) . Let I be an ideal of A and F a submodule of E such that IE ⊆ F . Then the following statement hold: (1) If I ⋉ F is a -absorbing δ ⋉ -primary ideal of A ⋉ E , then I is a -absorbing δ -primary ideal of A . (2) Assume that ( F : c ) = F for every c ∈ A \ I . Then I ⋉ F is a -absorbing δ ⋉ -primary ideal of A ⋉ E if and only if I is a -absorbing δ -primary ideal of A .Proof. (1) Assume that I ⋉ F is a 1-absorbing δ ⋉ -primary ideal of A ⋉ E andlet a, b, c be nonunit elements of A such that abc ∈ I . Thus ( a, b, c,
0) =( abc, ∈ I ⋉ F which implies that ( a, b, ∈ I ⋉ F or ( c, ∈ δ ⋉ ( I ⋉ F ) = δ ( I ) ⋉ E . Therefore ab ∈ I or c ∈ δ ( I ) and so (1) holds.(2) By (1), it suffices to prove the ”if” assertion. Let ( a, s ) , ( b, t ) , ( c, r ) benonunit elements of A ⋉ E such that ( a, s )( b, t )( c, r ) = ( abc, bcs + act + abr ) ∈ I ⋉ F . Clearly, abc ∈ I and so ab ∈ I or c ∈ δ ( I ) since I is a 1-absorbing δ -primary ideal of A . If c ∈ δ ( I ), then ( c, r ) ∈ δ ( I ) ⋉ E = δ ⋉ ( I ⋉ F ). Hence,we may assume that c / ∈ δ ( I ). Then ab ∈ I . As bcs + act + abr ∈ F and abr ∈ F , we get that bcs + act ∈ F . This implies bs + at ∈ ( F : c ) = F and so ( a, s )( b, t ) = ( ab, at + bs ) ∈ I ⋉ F . Therefore I ⋉ F is a 1-absorbing δ ⋉ -primary ideal of A ⋉ E . (cid:3) Corollary 2.31.
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