Generalization of 2-absorbing quasi primary ideals
GGENERALIZATION OF 2-ABSORBING QUASI PRIMARYIDEALS
EMEL ASLANKARAYI ˘GIT U ˘GURLU, ¨UNSAL TEKIR, AND SUAT KOC¸
Abstract.
In this article, we introduce and study the concept of φ -2-absorbingquasi primary ideals in commutative rings. Let R be a commutative ring witha nonzero identity and L ( R ) be the lattice of all ideals of R. Suppose that φ : L ( R ) → L ( R ) ∪ {∅} is a function. A proper ideal I of R is called a φ -2-absorbing quasi primary ideal of R if a, b, c ∈ R and whenever abc ∈ I − φ ( I ) , then either ab ∈ √ I or ac ∈ √ I or bc ∈ √ I. In addition to giving many prop-erties of φ -2-absorbing quasi primary ideals, we also use them to characterizevon Neumann regular rings.Keywords. weakly 2-absorbing quasi primary ideal, φ -2-absorbing quasiprimary ideal, von Neumann regular ring.2010 Mathematics Subject Classification. 13A15, 16E50 Introduction
In this article, we focus only on commutative rings with nonzero identity andnonzero unital modules. Let R always denote such a ring and M denote such an R -module. L ( R ) denotes the lattice of all ideals of R. Let I be a proper ideal of R , the set { r ∈ R | rs ∈ I for some s ∈ R (cid:31) I } will be denoted by Z I ( R ). Alsothe radical of I is defined as √ I := { r ∈ R | r k ∈ I for some k ∈ N } and for x ∈ R, ( I : x ) denote the ideal { r ∈ R | rx ∈ I } of R . A proper ideal I of acommutative ring R is prime if whenever a , a ∈ R with a a ∈ I , then a ∈ I or a ∈ I, [7]. In 2003, the authors [3] said that if whenever a , a ∈ R with0 R (cid:54) = a a ∈ I , then a ∈ I or a ∈ I, a proper ideal I of a commutative ring R is weakly prime . In [11], Bhatwadekar and Sharma defined a proper ideal I ofan integral domain R as almost prime (resp. n -almost prime) if for a , a ∈ R with a a ∈ I − I , (resp. a a ∈ I − I n , n ≥
3) then a ∈ I or a ∈ I . Thisdefinition can be made for any commutative ring R . Later, Anderson and Batanieh[1] introduced a concept which covers all the previous definitions in a commutativering R as following: Let φ : L ( R ) → L ( R ) ∪ {∅} be a function, where L ( R ) denotesthe set of all ideals of R . A proper ideal I of a commutative ring R is called φ - prime if for a , a ∈ R with a a ∈ I − φ ( I ), then a ∈ I or a ∈ I. They defined the map φ α : L ( R ) → L ( R ) ∪ {∅} as follows:(i) φ ∅ : φ ( I ) = ∅ defines prime ideals.(ii) φ : φ ( I ) = { R } defines weakly prime ideals.(iii) φ : φ ( I ) = I defines almost prime ideals.(iv) φ n : φ ( I ) = I n defines n -almost prime ideals( n ≥ . (v) φ ω : φ ( I ) = ∩ ∞ n =1 I n defines ω -prime ideals.(vi) φ : φ ( I ) = I defines any ideal. a r X i v : . [ m a t h . A C ] M a y EMEL ASLANKARAYI ˘GIT U ˘GURLU, ¨UNSAL TEKIR, AND SUAT KOC¸
The notion of 2-absorbing ideal, which is a generalization of prime ideal, wasintroduced by Badawi as the following: a proper ideal I of R is called a 2-absorbingideal of R if whenever a, b, c ∈ R and abc ∈ I , then ab ∈ I or ac ∈ I or bc ∈ I ,see [8]. Also, the notion is investigated in [5], [20], [12], [13] and [14]. Then, thenotion of 2-absorbing primary ideal, which is a generalization of primary ideal, wasintroduced in [10] as: a proper ideal I of R is called a 2-absorbing primary ideal of R if whenever a, b, c ∈ R and abc ∈ I , then ab ∈ I or ac ∈ √ I or bc ∈ √ I . Notethat a 2-absorbing ideal of a commutative ring R is a 2-absorbing primary ideal of R . But the converse is not true. For example, consider the ideal I = (20) of Z .Since 2 . . ∈ I , but 2 . / ∈ I and 2 . / ∈ I , I is not a 2-absorbing ideal of Z . However,it is clear that I is a 2-absorbing primary ideal of Z , since 2 . ∈ √ I . In 2016,the authors introduced the concept of φ -2-absorbing primary ideal which a properideal I of R is called a φ -2-absorbing primary ideal of R if whenever a, b, c ∈ R and abc ∈ I − φ ( I ), then ab ∈ I or ac ∈ √ I or bc ∈ √ I , see [9].On the other hand, the concept of quasi primary ideals in commutative rings wasintroduced and investigated by Fuchs in [16]. The author called an ideal I of R asa quasi primary ideal if √ I is a prime ideal. In [21], the notion of 2-absorbing quasiprimary ideal is introduced as following: a proper ideal I of R to be 2-absorbingquasi primary if √ I is a 2-absorbing ideal of R .In this paper, our aim to obtain the generalizations of the concept of the quasiprimary ideal and 2-absorbing quasi primary ideal. For this, firstly we define the φ -quasi primary ideal. Let φ : L ( R ) → L ( R ) ∪ {∅} be a function and I be a properideal of R. Then I is said to be a φ -quasi primary ideal if whenever a, b ∈ R and ab ∈ I − φ ( I ), then a ∈ √ I or b ∈ √ I . Similarly, I is called a φ -2-absorbing quasiprimary ideal if whenever a, b, c ∈ R and abc ∈ I − φ ( I ), then ab ∈ √ I or ac ∈ √ I or bc ∈ √ I . In Section 2, firstly we investigate the basic properties of a φ -quasi primaryideal and a φ -2-absorbing quasi primary. By the help of Theorem 1 and Theorem2, we give a diagram which clarifies the place of φ -2-absorbing quasi primary idealin the lattice of all ideals L ( R ) of R, see Figure 1. In Proposition 4, we give amethod for constructing φ -2-absorbing quasi primary ideal in commutative rings.Also, if φ ( I ) is a quasi primary ideal of R, we prove that I is a φ -2-absorbing quasiprimary ideal of R ⇔ I is a 2-absorbing quasi primary ideal of R, see Corollary3. With Theorem 5, we obtain the Nakayama´s Lemma for weakly (2-absorbing)quasi primary ideals. Moreover, we examine the notion of ” φ -2-absorbing quasiprimary ideal” in S − R, where S is a multiplicatively subset of R . In Theorem7, we characterize the weakly 2-absorbing quasi primary ideal of R ∝ M, that is,the trivial ring extension, where M is an R -module. In Theorem 8, we describevon Neumann regular rings in terms of φ -2-absorbing quasi primary ideals. Finally,with the all results of the Section 3, we characterize φ -2-absorbing quasi primaryideal in direct product of finitely many commutative rings.ENERALIZATION OF 2-ABSORBING QUASI PRIMARY IDEALS Characterization of φ -2-absorbing quasi primary ideals Throughout the paper, φ : L ( R ) → L ( R ) ∪ {∅} always denotes a function. Definition 1.
Let R be a ring and I a proper ideal of R. (i) I is said to be a φ -quasi primary ideal if whenever a, b ∈ R and ab ∈ I − φ ( I ) ,then a ∈ √ I or b ∈ √ I .(ii) I is said to be a φ -2-absorbing quasi primary ideal if whenever a, b, c ∈ R and abc ∈ I − φ ( I ) , then ab ∈ √ I or ac ∈ √ I or bc ∈ √ I . Definition 2.
Let φ α : L ( R ) → L ( R ) ∪{∅} be one of the following special functionsand I a φ α -quasi primary ( φ α -2-absorbing quasi primary) ideal of R. Then, φ ∅ ( I ) = ∅ quasi primary (2-absorbing quasi primary) ideal φ ( I ) = 0 R weakly quasi primary (weakly 2-absorbing quasi primary) ideal φ ( I ) = I almost quasi primary (almost 2-absorbing quasi primary) ideal φ n ( I ) = I n n -almost quasi primary ( n -almost 2-absorbing quasi primary) ideal ( n ≥ φ ω ( I ) = ∩ ∞ n =1 I n ω -quasi primary ( ω -2-absorbing quasi primary) ideal φ ( I ) = I any ideal. Note that since I − φ ( I ) = I − ( I ∩ φ ( I )), for any ideal I of R , without loss ofgenerality, assume that φ ( I ) ⊆ I. Let ψ , ψ : L ( R ) → L ( R ) ∪ {∅} be two functions,if ψ ( I ) ⊆ ψ ( I ) for each I ∈ L ( R ) , we denote ψ ≤ ψ . Thus clearly, we have thefollowing order: φ ∅ ≤ φ ≤ φ ω ≤ · · · ≤ φ n +1 ≤ φ n ≤ · · · ≤ φ ≤ φ . Also 2-almostquasi primary (2-almost 2-absorbing quasi primary) ideals are exactly almost quasiprimary (almost 2-absorbing quasi primary) ideals. Proposition 1.
Let R be a ring and I be a proper ideal R. Let ψ , ψ : L ( R ) → L ( R ) ∪ {∅} be two functions with ψ ≤ ψ . (i) If I is a ψ -quasi primary ideal of R, then I is a ψ -quasi primary ideal of R. (ii) I is a quasi primary ideal ⇒ I is a weakly quasi primary ideal ⇒ I is an ω -quasi primary ideal ⇒ I is an ( n + 1) -almost quasi primary ideal ⇒ I is an n -almost quasi primary ideal ( n ≥ ⇒ I is an almost quasi primary ideal.(iii) I is an ω -quasi primary ideal if and only if I is an n -almost quasi primaryideal for each n ≥ . (iv) If I is a ψ -2-absorbing quasi primary ideal of R, then I is a ψ -2-absorbingquasi primary ideal of R. (v) I is 2-absorbing quasi primary ideal ⇒ I is weakly 2-absorbing quasi primaryideal ⇒ I is an ω -2-absorbing quasi primary ideal ⇒ I is an ( n + 1) -almost 2-absorbing quasi primary ideal ⇒ I is an n -almost 2-absorbing quasi primary ideal ( n ≥ ⇒ I is an almost 2-absorbing quasi primary ideal.(vi) I is an ω -2-absorbing quasi primary ideal if and only if I is an n -almost2-absorbing quasi primary ideal for each n ≥ . Proof. (i): It is evident.(ii): Follows from (i).(iii): Every ω -quasi primary ideal is an n -almost quasi primary ideal for each n ≥ φ ω ≤ φ n . Now, let I be an n -almost quasi primary ideal for each n ≥ . Choose elements a, b ∈ R such that ab ∈ I − ∩ ∞ n =1 I n . Then we have ab ∈ EMEL ASLANKARAYI ˘GIT U ˘GURLU, ¨UNSAL TEKIR, AND SUAT KOC¸ I − I n for some n ≥ . Since I is an n -almost quasi primary ideal of R, we concludeeither a ∈ √ I or b ∈ √ I. Therefore, I is an ω -quasi primary ideal.(iv): It is evident.(v): Follows from (iv).(vi): Similar to (iii). (cid:3) Theorem 1. (i) If √ I = I, then I is a φ -2-absorbing quasi primary ideal of R ifand only if I is a φ -2-absorbing ideal of R. (ii) If I is a φ -2-absorbing quasi primary ideal of R and φ ( √ I ) = (cid:112) φ ( I ) , then √ I is a φ -2-absorbing ideal of R .(iii) If √ I is a φ -2-absorbing ideal of R and φ ( √ I ) ⊆ φ ( I ) , then I is a φ -2-absorbing quasi primary ideal of R. (iv) If I is a φ -quasi primary ideal of R and φ ( √ I ) = (cid:112) φ ( I ) , then √ I is a φ -prime ideal of R .(v) If √ I is a φ -prime ideal of R and φ ( √ I ) ⊆ φ ( I ) , then I is a φ -quasi primaryideal of R. Proof. (i): It is evident.(ii): Let I be a φ -2-absorbing quasi primary ideal of R. Take a, b, c ∈ R suchthat abc ∈ √ I − φ ( √ I ). Then there exists a positive integer n such that ( abc ) n = a n b n c n ∈ I. Since abc / ∈ φ ( √ I ) and φ ( √ I ) = (cid:112) φ ( I ) , we get abc / ∈ (cid:112) φ ( I ) , so a n b n c n / ∈ φ ( I ) . Thus, by our hypothesis, a n b n ∈ √ I or b n c n ∈ √ I or a n c n ∈ √ I .Consequently, ab ∈ √ I or bc ∈ √ I or ac ∈ √ I .(iii): Assume that √ I is a φ -2-absorbing ideal of R. Choose a, b, c ∈ R such that abc ∈ I − φ ( I ). Since I ⊆ √ I and φ ( √ I ) ⊆ φ ( I ) , we have abc ∈ √ I − φ ( √ I ) . Thenas √ I is φ -2-absorbing, ab ∈ √ I or bc ∈ √ I or ac ∈ √ I . So I is a φ -quasi primaryideal of R. (iv): It is similar to (i).(v): It is similar to (ii). (cid:3) Theorem 2. (i) Every φ -quasi primary ideal is a φ -2-absorbing primary ideal.(ii) Every φ -2-absorbing primary ideal is a φ -2-absorbing quasi primary ideal.(iii) Every φ -quasi primary ideal is a φ -2-absorbing quasi primary ideal.Proof. (i): Let I be a φ -quasi primary ideal and choose a, b, c ∈ R such that abc = a ( bc ) ∈ I − φ ( I ) . Since I is a φ -quasi primary ideal, we conclude either a ∈ √ I or bc ∈ √ I .
Then we have either ac ∈ √ I or bc ∈ √ I which completes the proof.(ii): It is clear.(iii): It follows from (i) and (ii). (cid:3) By Theorem 1 and Theorem 2, we give the following diagram which clarifies theplace of φ -2-absorbing quasi primary ideal in the lattice of all ideals L ( R ) of R. ENERALIZATION OF 2-ABSORBING QUASI PRIMARY IDEALS Figure 1. φ -2-absorbing quasi primary ideal vs other classical φ -ideals Corollary 1. If I is a φ -2-absorbing primary ideal of R and φ ( √ I ) = (cid:112) φ ( I ) , then √ I is a φ -2-absorbing ideal of R .Proof. It follows from Theorem 1(ii) and Theorem 2(ii). (cid:3)
Proposition 2.
Let I be a proper ideal of R. Then,(i) I is a φ -quasi primary ideal of R if and only if I/φ ( I ) is a weakly quasiprimary ideal of R/φ ( I ) . (ii) I is a φ -2-absorbing quasi primary ideal of R if and only I/φ ( I ) is a weakly2-absorbing quasi primary ideal of R/φ ( I ) . Proof. (i): Suppose that I is a φ -quasi primary ideal of R. Let 0
R/φ ( I ) (cid:54) = ( a + φ ( I ))( b + φ ( I )) = ab + φ ( I ) ∈ I/φ ( I ) for some a, b ∈ R. Then we have ab ∈ I − φ ( I ) . Since I is a φ -quasi primary ideal of R, we conclude either a ∈ √ I or b ∈ √ I. This implies that a + φ ( I ) ∈ √ I/φ ( I ) = (cid:112) I/φ ( I ) or b + φ ( I ) ∈ (cid:112) I/φ ( I ).Therefore, I/φ ( I )is a weakly quasi primary ideal of R/φ ( I ) . Conversely, assumethat
I/φ ( I ) is a weakly quasi primary ideal of R/φ ( I ) . Now, choose a, b ∈ R suchthat ab ∈ I − φ ( I ) . This yields that 0
R/φ ( I ) (cid:54) = ( a + φ ( I ))( b + φ ( I )) = ab + φ ( I ) ∈ I/φ ( I ) . Since
I/φ ( I ) is a weakly quasi primary ideal of R/φ ( I ) , we get either a + φ ( I ) ∈ (cid:112) I/φ ( I ) = √ I/φ ( I ) or b + φ ( I ) ∈ √ I/φ ( I ). Then we have a ∈ √ I or b ∈ √ I. Hence, I is a φ -quasi primary ideal of R. (ii): Similar to (i). (cid:3) In the following, we characterize all quasi primary and 2-absorbing quasi pri-mary ideals in factor rings
R/φ ( I ) . Since the proof is similar to that of previousproposition (i), we omit the proof.
Proposition 3.
Let I be a proper ideal of R. Then,
EMEL ASLANKARAYI ˘GIT U ˘GURLU, ¨UNSAL TEKIR, AND SUAT KOC¸ (i) I is a quasi primary ideal of R if and only if I/φ ( I ) is a quasi primary idealof R/φ ( I ) . (ii) I is a 2-absorbing quasi primary ideal of R if and only I/φ ( I ) is a 2-absorbingquasi primary ideal of R/φ ( I ) . Now, we give a method for constructing φ -2-absorbing quasi primary ideal incommutative rings. Proposition 4.
Let P , P be φ -quasi primary ideal of a ring R. Then the followingstatements hold:(i) If φ ( P ) = φ ( P ) = φ ( P ∩ P ) , then P ∩ P is a φ -2-absorbing quasi primaryideal of R. (ii) If φ ( P ) = φ ( P ) = φ ( P P ) , then P P is a φ -2-absorbing quasi primaryideal of R Proof. (i): Let abc ∈ P ∩ P − φ ( P ∩ P ) for some a, b, c ∈ R. Then we have abc ∈ P − φ ( P ) . As P is a φ -quasi primary ideal, we conclude either a ∈ √ P or b ∈ √ P or c ∈ √ P . Similarly, we get either a ∈ √ P or b ∈ √ P or c ∈√ P . Without loss generality, we may assume that a ∈ √ P and b ∈ √ P . Then ab ∈ √ P √ P ⊆ √ P ∩ √ P = √ P ∩ P . Hence, P ∩ P is a φ -2-absorbing quasiprimary ideal of R. (ii): Similar to (i). (cid:3) Definition 3.
Let I be a φ -2-absorbing quasi primary ideal of R and a, b, c ∈ R suchthat abc ∈ φ ( I ) . If ab / ∈ √ I, ac / ∈ √ I and bc / ∈ √ I, then ( a, b, c ) is called a strongly- φ -triple zero of I. In particular, if φ ( I ) = 0 , then ( a, b, c ) is called a strongly-triplezero of I. Remark 1. If I is a φ -2-absorbing quasi primary ideal of R that is not a 2-absorbing quasi primary ideal, then I has a strongly- φ -triple zero ( a, b, c ) for some a, b, c ∈ R. Proposition 5.
Suppose that I is a weakly 2-absorbing quasi primary ideal of R which is not 2-absorbing quasi primary ideal, then I = 0 . Proof.
Let I be a weakly 2-absorbing quasi primary ideal of R such that I (cid:54) =0 . Now, we will show that I is a 2-absorbing quasi primary ideal of R. Choose a, b, c ∈ R such that abc ∈ I. Since I is weakly 2-absorbing quasi primary ideal,we may assume that abc = 0 . Otherwise, we would have ab ∈ √ I or bc ∈ √ I or ac ∈ √ I. If abI (cid:54) = 0, then there exists x ∈ I such that abx (cid:54) = 0 . Since 0 (cid:54) = abx = ab ( c + x ) ∈ I and I is weakly 2-absorbing quasi primary ideal, we get either ab ∈ √ I or a ( c + x ) ∈ √ I or b ( c + x ) ∈ √ I. Thus we have either ab ∈ √ I or ac ∈ √ I or bc ∈ √ I, then we are done. So assume that abI = 0 = acI = bcI. On the otherhand, if aI (cid:54) = 0 , then there exists x , x ∈ I such that ax x (cid:54) = 0 . Then we have0 (cid:54) = a ( b + x )( c + x ) = ax x ∈ I since abI = acI = 0 . As I is a weakly 2-absorbing quasi primary ideal, we get either a ( b + x ) ∈ √ I or a ( c + x ) ∈ √ I or( b + x )( c + x ) ∈ √ I. Then we have ab ∈ √ I or bc ∈ √ I or ac ∈ √ I. So assumethat aI = 0 . Similarly, we may assume that bI = cI = 0 . As I (cid:54) = 0 , there exist y, z, w ∈ I such that yzw (cid:54) = 0 . As abI = 0 = acI = bcI = aI = bI = cI , it isclear that 0 (cid:54) = yzw = ( a + y )( b + z )( c + w ) ∈ I. This implies that ( a + y )( b + z ) ∈ √ I or ( a + y )( c + w ) ∈ √ I or ( b + z )( c + w ) ∈ √ I and so we have ab ∈ √ I or bc ∈ √ I or ac ∈ √ I. Hence, I is a 2-absorbing quasi primary ideal of R. (cid:3) ENERALIZATION OF 2-ABSORBING QUASI PRIMARY IDEALS Corollary 2. If I is a weakly 2-absorbing quasi primary ideal of R which is not2-absorbing quasi primary ideal, then √ I = √ . Theorem 3. (i) Let I be a φ -2-absorbing quasi primary ideal of R. Then either I is a 2-absorbing quasi primary ideal or I ⊆ φ ( I ) . (ii) If I is a φ -2-absorbing quasi primary ideal of R which is not 2-absorbingquasi primary ideal, then √ I = (cid:112) φ ( I ) . Proof. (i) Suppose that I is a φ -2-absorbing quasi primary ideal of R that is not2-absorbing quasi primary ideal. Then note that I/φ ( I ) is not a 2-absorbing quasiprimary ideal of R/φ ( I ) . Also by Proposition 2,
I/φ ( I ) is a weakly 2-absorbing quasiprimary ideal of R/φ ( I ) . Then by Proposition 5, we get (
I/φ ( I )) = 0 R/φ ( I ) andthis yields I ⊆ φ ( I ) . (ii): Suppose that I is a φ -2-absorbing quasi primary ideal of R which is not2-absorbing quasi primary ideal. Then by (i), we have I ⊆ φ ( I ) and thus √ I ⊆ (cid:112) φ ( I ) . On the other hand, since φ ( I ) ⊆ I, we have √ I = (cid:112) φ ( I ) . (cid:3) Corollary 3.
Suppose that I is a proper ideal of R such that φ ( I ) is a quasi primaryideal of R. Then the following statments are equivalent:(i) I is a φ -2-absorbing quasi primary ideal of R. (ii) I is a 2-absorbing quasi primary ideal of R. Proof. ( i ) ⇒ ( ii ) : Suppose that I is a φ -2-absorbing quasi primary ideal of R. Now,we will show that I is a 2-absorbing quasi primary ideal of R. Deny. Then Theorem 3(ii), we have √ I = (cid:112) φ ( I ) . Since φ ( I ) is a quasi primary ideal, we have √ I = (cid:112) φ ( I )is a prime ideal and so I is quasi primary. Then by [21, Proposition 2.6], I is a2-absorbing quasi primary, a contradiction.( ii ) ⇒ ( i ) : Directly from definition. (cid:3) Theorem 4. (i) If P is a weakly quasi primary ideal of R that is not quasi primary,then P = 0 . (ii) If P is a φ -quasi primary ideal of R that is not quasi primary, then P ⊆ φ ( P ) . (iii) If P is a φ -quasi primary ideal of R where φ ≤ φ , then P is n -almost quasiprimary for all n ≥ , so ω -quasi primary.Proof. (i): Similar to Proposition 5.(ii): Similar to Theorem 3 (i).(ii): Assume that P is a φ -quasi primary ideal of R and φ ≤ φ . If P is quasiprimary, then P is φ -quasi primary for each φ . If P is not quasi primary, by (i), P ⊆ φ ( P ) . Also as φ ≤ φ , we get P ⊆ φ ( P ) ⊆ P , so φ ( P ) = P n for each n ≥ . Consequently, since P is φ -quasi primary, P is n -almost quasi primary for all n ≥ , so ω -quasi primary by Proposition 1 (iii). (cid:3) Now, we give the Nakayama’s Lemma for weakly (2-absorbing) quasi primaryideals as follows.
Theorem 5. (Nakayama’s Lemma) Let P be a weakly 2-absorbing quasi primary(weakly quasi primary) ideal of R that is not 2-absorbing quasi primary (quasiprimary) and M be an R -module. Then the following statements hold:(i) P ⊆ Jac ( R ) , where Jac ( R ) is the jacobson radical of R. (ii) If P M = M, then M = 0 . (iii) If N is a submodule of M such that P M + N = M, then N = M. EMEL ASLANKARAYI ˘GIT U ˘GURLU, ¨UNSAL TEKIR, AND SUAT KOC¸
Proof. ( i ) : Suppose that P is a weakly 2-absorbing quasi primary ideal of R thatis not 2-absorbing quasi primary . Then by Theorem 5, P = 0. Let x ∈ P. Now,we will show that 1 − rx is a unit of R for each r ∈ R. Note that rx ∈ P andso r x = 0 . This implies that 1 = 1 − r x = (1 − rx )(1 + rx + r x ) . Thus x ∈ Jac ( R ) and so P ⊆ Jac ( R ) . ( ii ) : Suppose that P M = M. Then by Theorem 5, P = 0 and so M = P M = P M = 0 . ( iii ) : Follows from (ii). (cid:3) Theorem 6.
Let S be a multiplicatively closed subset of R and φ q : L ( S − R ) → L ( S − R ) ∪ {∅} , defined by φ q ( S − I ) = S − ( φ ( I )) for each ideal I of R, be a func-tion. Then the following statements hold:(i) If P is a φ -2-absorbing quasi primary ideal of R with S ∩ P = ∅ , then S − P is a φ q -2-absorbing quasi primary ideal of S − R. (ii) Let P be an ideal of R such that Z φ ( P ) ( R ) ∩ S = ∅ and Z P ( R ) ∩ S = ∅ . If S − P is a φ q -2-absorbing quasi primary ideal of S − R, then P is a φ -2-absorbingquasi primary ideal of R. Proof. (i): Let as bt cu ∈ S − P − φ q ( S − P ) for any a, b, c ∈ R and s, t, u ∈ S. As φ q ( S − P ) = S − ( φ ( P )) , we get t ∗ abc = ( t ∗ a ) bc ∈ P − φ ( P ) for some t ∗ ∈ S. Since P is a φ -2-absorbing quasi primary ideal of R, we get t ∗ ab ∈ √ P or t ∗ ac ∈ √ P or bc ∈ √ P .
This implies that abst = t ∗ abt ∗ st ∈ S − √ P = √ S − P or acsu = t ∗ act ∗ su ∈ √ S − P or bctu ∈ √ S − P .
Hence S − P is a φ q -2-absorbing quasi primary ideal of S − R. (ii): Let abc ∈ P − φ ( P ) for some a, b, c ∈ R. Then a b c ∈ S − P. Since Z φ ( P ) ( R ) ∩ S = ∅ , it is clear that a b c / ∈ S − ( φ ( P )) = φ q ( S − P ) . As S − P is a φ q -2-absorbingquasi primary ideal of S − R, we get either a b ∈ √ S − P = S − √ P or a c ∈ S − √ P or b c ∈ S − √ P .
Without loss generality, we may assume that b c ∈ S − √ P .
Thenwe get tbc ∈ √ P and so t n b n c n ∈ P for some n ∈ N . If b n c n / ∈ P, then we have t n ∈ Z P ( R ) ∩ S, a contradiction. So we have b n c n ∈ P and thus bc ∈ √ P .
Thus P is a φ -2-absorbing quasi primary ideal of R. (cid:3) Let M be an R -module. The trivial ring extension (or idealization) R ∝ M = R ⊕ M of M is a commutative ring with componentwise addition and multiplication( a, m )( b, m (cid:48) ) = ( ab, am (cid:48) + bm ) for each a, b ∈ R ; m, m (cid:48) ∈ M [17]. Let I be an idealof R and N is a submodule of M. Then I ∝ N = I ⊕ N is an ideal of R ∝ M if and only if IM ⊆ N [4] . In that case, I ∝ N is called a homogeneous ideal of R ∝ M. For any ideal I ∝ N of R ∝ M, the radical of I ∝ N is characterized asfollows: √ I ∝ N = √ I ∝ M [4, Theorem 3.2].Now, we characterize certain weakly 2-absorbing quasi primary ideals in trivialring extensions. Theorem 7.
Let M be an R -module and I a proper ideal of R. Then the followingstatements are equivalent:(i) I ∝ M is a weakly 2-absorbing quasi primary ideal of R ∝ M. (ii) I is a weakly 2-absorbing quasi primary ideal of R and for any strongly triplezero ( a, b, c ) of I, we have abM = 0 = acM = bcM. ENERALIZATION OF 2-ABSORBING QUASI PRIMARY IDEALS Proof. ( i ) ⇒ ( ii ) : Suppose that I ∝ M is a weakly 2-absorbing quasi primary idealof R ∝ M. Now, we will show that I is a weakly 2-absorbing quasi primary ideal of R. Let 0 (cid:54) = abc ∈ I. Then note that (0 , M ) (cid:54) = ( a, M )( b, M )( c, M ) = ( abc, M ) ∈ I ∝ M. As I ∝ M is a weakly 2-absorbing quasi primary ideal, we concludeeither ( a, M )( b, M ) = ( ab, M ) ∈ √ I ∝ M = √ I ∝ M or ( ac, M ) ∈ √ I ∝ M or( bc, M ) ∈ √ I ∝ M. This implies that ab ∈ √ I or ac ∈ √ I or bc ∈ √ I. Therefore, I is a weakly 2-absorbing quasi primary ideal of R. Let ( x, y, z ) be a strongly triplezero of I. Then xyz = 0 and also xy, xz, yz / ∈ √ I. Assume that xyM (cid:54) = 0 . Then thereexists m ∈ M such that xym (cid:54) = 0 . Then note that (0 , M ) (cid:54) = ( x, M )( y, M )( z, m ) =(0 , xym ) ∈ I ∝ M. Since I ∝ M is a weakly 2-absorbing quasi primary ideal, weconclude either ( x, M )( y, M ) = ( xy, M ) ∈ √ I ∝ M = √ I ∝ M or ( xz, xm ) ∈√ I ∝ M or ( yz, ym ) ∈ √ I ∝ M, a contradiction. Thus xM = yM = zM = 0 . ( ii ) ⇒ ( i ) : Suppose that (0 , M ) (cid:54) = ( a, m )( b, m (cid:48) )( c, m (cid:48)(cid:48) ) = ( abc, abm (cid:48)(cid:48) + acm (cid:48) + bcm ) ∈ I ∝ M for some a, b, c ∈ R ; m, m (cid:48) , m (cid:48)(cid:48) ∈ M. Then abc ∈ I. If abc (cid:54) = 0 , theneither ab ∈ √ I or ac ∈ √ I or bc ∈ √ I. This implies that ( a, m )( b, m (cid:48) ) ∈ √ I ∝ M = √ I ∝ M or ( a, m )( c, m (cid:48)(cid:48) ) ∈ √ I ∝ M or ( b, m (cid:48) )( c, m (cid:48)(cid:48) ) ∈ √ I ∝ M . Now assumethat abc = 0 . If ( a, b, c ) is a strongly triple zero of I, then by assumption abM =0 = acM = bcM and so (0 , M ) = ( abc, abm (cid:48)(cid:48) + acm (cid:48) + bcm ) = ( a, m )( b, m (cid:48) )( c, m (cid:48)(cid:48) )which is a contradiction. So that ( a, b, c ) is not strongly triple zero and this yields ab ∈ √ I or ac ∈ √ I or bc ∈ √ I. Therefore, we have ( a, m )( b, m (cid:48) ) ∈ √ I ∝ M or( a, m )( c, m (cid:48)(cid:48) ) ∈ √ I ∝ M or ( b, m (cid:48) )( c, m (cid:48)(cid:48) ) ∈ √ I ∝ M .
Hence, I ∝ M is a weakly2-absorbing quasi primary ideal of R ∝ M. (cid:3) Let R , R , . . . , R n be commutative rings and R = R × R × · · · × R n be thedirect product of those rings. It is well known that every ideal of R has the form I = I × I × · · · × I n , where I k is an ideal of R k for each 1 ≤ k ≤ n. Supposethat ψ i : L ( R i ) → L ( R i ) ∪ {∅} is a function for each 1 ≤ i ≤ n and put φ := ψ × ψ × · · · × ψ n , that is, φ ( I ) = ψ ( I ) × ψ ( I ) × · · · × ψ n ( I n ) . Then note that φ : L ( R ) → L ( R ) ∪ {∅} becomes a function.Recall that a commutative ring R is said to be a von Neumann regular ring if for each a ∈ R, there exists x ∈ R such that a = a x [22] . In this case, theprincipal ideal ( a ) is a generated by an idempotent element e ∈ R. The notion ofvon Neumann regular ring has an important place in commutative algebra. So far,there have been many generalizations of this concept. See, for example, [19], [2]and [6]. Now, we characterize von Neumann regular rings in terms of φ -2-absorbingquasi primary ideals. Theorem 8.
Let R , R , . . . , R m be commutative rings and R = R × R × · · · × R m , where ≤ m < ∞ . Suppose that n ≥ . Then the following statements areequivalent.(i) Every ideal of R is a φ n -2-absorbing quasi primary ideal.(ii) R , R , . . . , R m are von Neumann regular rings.Proof. ( i ) ⇒ ( ii ) : Suppose that every ideal of R is a φ n -2-absorbing quasi primaryideal. Now, we will show that R , R , . . . , R m are von Neumann regular rings.Suppose not. Without loss of generality, we may assume that R is not a vonNeumann regular ring. Then there exists an ideal I of R such that I n (cid:32) I . Thenwe can find an element a ∈ I − I n . Now, put J = I × × × R × R ×· · ·× R m andalso x = ( a, , , , . . . , , x = (1 , , , , . . . , , x = (1 , , , , . . . , . Then notethat x x x ∈ J − φ n ( J ) . As x x , x x and x x / ∈ √ J, J is not a φ n -2-absorbing quasi primary ideal of R which is a contradiction. Thus R , R , . . . , R m are vonNeumann regular rings.( ii ) ⇒ ( i ) : Suppose that R , R , . . . , R m are von Neumann regular rings. Thennote that I ni = I i for any ideal I i of R i . Take any ideal J of R. Then J = J × J ×· · ·× J m for some ideal J k of R k , where 1 ≤ k ≤ m. Then J n = J n × J n ×· · ·× J nm = φ n ( J ) = J × J ×· · ·× J m = J. This implies that J − φ n ( J ) = ∅ and so J is trivially φ n -2-absorbing quasi primary ideal of R. (cid:3) Corollary 4.
Let R be a ring and n ≥ . Then the following statements are equiv-alent:(i) Every ideal of R is a φ n -2-absorbing quasi primary ideal.(ii) R is a von Neumann regular ring. φ -(2-absorbing) quasi primary ideals in direct product of rings In this section, we investigate φ -2-absorbing quasi primary ideal in direct productof finitely many commutative rings. Theorem 9.
Let R = R × R , where R and R are rings and ψ i : L ( R i ) → L ( R i ) ∪ {∅} is a function for each i = 1 , . Let φ := ψ × ψ and J an ideal of R. Then J is a φ -quasi primary ideal of R if and only if J has exactly one of thefollowing three types:(i) J = I × I such that ψ i ( I i ) = I i for i = 1 , .(ii) I × R for some ψ -quasi primary ideal I of R which must be quasi primaryif ψ ( R ) (cid:54) = R .(iii) R × I for some ψ -quasi primary ideal I of R which must be quasiprimary if ψ ( R ) (cid:54) = R . Proof. ⇒ : Suppose that J is a φ -quasi primary ideal of R. Then J = I × I forsome ideal I of R and some ideal I of R . Let xy ∈ I − ψ ( I ) . Then we have( x, y,
0) = ( xy, ∈ J − φ ( J ) . As J is a φ -quasi primary ideal, we concludeeither ( x, ∈ √ J or ( y, ∈ √ J. Since √ J = √ I × √ I , we get x ∈ √ I or y ∈ √ I . Hence, I is a ψ -quasi primary ideal. Similarly, I is a ψ -quasiprimary ideal. We may assume that J (cid:54) = φ ( J ) . Then we have either I (cid:54) = ψ ( I )or I (cid:54) = ψ ( I ) . Without loss of generality, we may assume that I (cid:54) = ψ ( I ) . Sothere exists a ∈ I − ψ ( I ) . Take b ∈ I . Then we have ( a, , b ) ∈ J − φ ( J ) . Thisimplies either ( a, ∈ √ J or (1 , b ) ∈ √ J. Then we get 1 ∈ √ I or 1 ∈ √ I , that is, I = R or I = R . Now, assume that I = R . Now, we will show that I is a quasiprimary ideal provided that ψ ( R ) (cid:54) = R . So suppose ψ ( R ) (cid:54) = R . Let xy ∈ I forsome x, y ∈ R . Then we have ( x, y,
1) = ( xy, ∈ I × R − φ ( I × R ) . As J isa φ -quasi primary ideal, we get either ( x, ∈ √ J or ( y, ∈ √ J. Hence, x ∈ √ I or y ∈ √ I . Therefore, I is a quasi primary ideal. ⇐ : Suppose that J = I × I such that ψ i ( I i ) = I i for i = 1 ,
2. Since φ ( I × I ) = ψ ( I ) × ψ ( I ) = I × I , we get I × I − φ ( I × I ) = ∅ and so J is trivially φ -quasiprimary ideal. Let J = I × R for some ψ -quasi primary ideal I of R which mustbe quasi primary if ψ ( R ) (cid:54) = R . First, assume that ψ ( R ) = R . Then note that φ ( J ) = ψ ( I ) × R . Let ( x , x )( y , y ) = ( x y , x y ) ∈ J − φ ( J ) for some x i , y i ∈ R i . Then we have x y ∈ I − ψ ( I ) . This yields that x ∈ √ I or y ∈ √ I since I is a ψ -quasi primary ideal. Then we get either ( x , x ) ∈ √ I × R = √ I × R or ( y , y ) ∈ √ I × R . Hence, J is a φ -quasi primary ideal of R. Now, assume that ψ ( R ) (cid:54) = R and I is a quasi primary ideal. Then I × R is a quasi primary idealENERALIZATION OF 2-ABSORBING QUASI PRIMARY IDEALS of R by [21, Lemma 2.2]. Hence, J = I × R is a φ -quasi primary ideal of R. Inother case, one can see that J is a φ -quasi primary ideal of R. (cid:3) Theorem 10.
Let R = R × R × · · · × R n , where R , R , . . . , R n are rings and ψ i : L ( R i ) → L ( R i ) ∪{∅} be a function for each i = 1 , , . . . , n . Let φ := ψ × ψ ×· · ·× ψ n and J be an ideal of R. Then J is a φ -quasi primary ideal of R if and only if J hasexactly one of the following two types:(i) J = I × I × · · · × I n such that ψ i ( I i ) = I i for i = 1 , , . . . , n .(ii) J = R × R × · · · × R t − × I t × R t +1 × · · · × R n for some ψ t -quasi primaryideal I t of R t which must be quasi primary if ψ j ( R j ) (cid:54) = R j for some j (cid:54) = t. Proof.
We use induction on n to prove the claim. If n = 1 , the claim is clear. If n = 2 , the claim follows from previous theorem. Assume that the claim is true for all n < k and put n = k. Put R (cid:48) = R × R × · · · × R k − , J (cid:48) = I × I × · · · × I k − and φ (cid:48) = ψ × ψ × · · · × ψ k − . Then note that R = R (cid:48) × R k , J = J (cid:48) × J k and φ = φ (cid:48) × ψ k . Then by previous theorem, J is a φ -quasi primary ideal of R if andonly if one of the following conditions hold: (i) J = J (cid:48) × I k such that φ (cid:48) ( J (cid:48) ) = J (cid:48) and ψ k ( I k ) = I k (ii) J = J (cid:48) × R k for some φ (cid:48) -quasi primary ideal J (cid:48) of R (cid:48) whichmust be quasi primary if ψ k ( R k ) (cid:54) = R k (iii) J = R (cid:48) × I k for some ψ k -quasi primaryideal I k of R k which must be quasi primary if φ (cid:48) ( R (cid:48) ) (cid:54) = R (cid:48) . The rest follows frominduction hypothesis and [21, Theorem 2.3]. (cid:3)
Theorem 11.
Let R and R be commutative rings with identity and let R = R × R . Suppose that ψ i : L ( R i ) → L ( R i ) ∪ {∅} ( i = 1 , are functions such that ψ ( R ) (cid:54) = R and φ = ψ × ψ . Then the following assertions are equivalent:(i) I × R is a φ -2-absorbing quasi primary ideal of R. (ii) I × R is a 2-absorbing quasi primary ideal of R. (iii) I is a 2-absorbing quasi primary ideal of R . Proof.
Assume that ψ ( I ) = ∅ or ψ ( R ) = ∅ . Then clearly φ ( I × R ) = ∅ so that( i ) ⇔ ( ii ) ⇔ ( iii ) follows from [21, Theorem 2.23]. Hence suppose that ψ ( I ) (cid:54) = ∅ and ψ ( R ) (cid:54) = ∅ , so φ ( I × R ) (cid:54) = ∅ . ( i ) ⇒ ( ii ) : Suppose that I × R is a φ -2-absorbing quasi primary ideal of R. Similar argument in the proof of Theorem 9 shows that I is a ψ -2-absorbingquasi primary ideal of R . If I is 2-absorbing quasi primary, then I × R is a 2-absorbing quasi primary ideal of R, by [21, Theorem 2.23]. If I is not 2-absorbingquasi primary, then I has a strongly ψ -triple zero ( x, y, z ) for some x, y, z ∈ R by Remark 1. Then ( x, y, z,
1) = ( xyz, ∈ I × R − ψ ( I ) × ψ ( R ) since ψ ( R ) (cid:54) = R . This implies that xy ∈ √ I or yz ∈ √ I or xz ∈ √ I , a contradiction.Thus I is 2-absorbing quasi primary. Consequently, I × R is a 2-absorbing quasiprimary ideal of R. ( ii ) ⇒ ( iii ) and ( iii ) ⇒ ( i ) : Follows from [21, Theorem 2.23]. (cid:3) Theorem 12.
Let R and R be commutative rings with identity and let R = R × R . Suppose that ψ i : S ( R i ) → S ( R i ) ∪ {∅} ( i = 1 , are functions and φ = ψ × ψ . The following statements are equivalent:(i) I × R is a φ -2-absorbing quasi primary ideal of R that is not a 2-absorbingquasi primary ideal of R. (ii) φ ( I × R ) (cid:54) = ∅ , ψ ( R ) = R and I is a ψ -2-absorbing quasi primary idealof R that is not a 2-absorbing quasi primary ideal of R . Proof. ( i ) ⇒ ( ii ) : Let I × R be φ -2-absorbing quasi primary that is not 2-absorbing quasi primary. By Theorem 11, since I × R is not a 2-absorbing quasiprimary ideal of R, one can see that φ ( I × R ) (cid:54) = ∅ and ψ ( R ) = R . As I × R is a φ -2-absorbing quasi primary ideal of R, it is clear that I is a ψ -2-absorbingquasi primary ideal of R . Also, since I × R is not a 2-absorbing quasi primaryideal of R, I is not a 2-absorbing quasi primary ideal of R by [21, Theorem 2.3].( ii ) ⇒ ( i ) : Since φ ( I × R ) (cid:54) = ∅ and ψ ( R ) = R , we get R/φ ( I × R ) ∼ = R /ψ ( R ) and I × R /φ ( I × R ) ∼ = I /ψ ( I ) . By Proposition 2(ii), since I is a ψ -2-absorbing quasi primary ideal of R , I /ψ ( I ) is a weakly 2-absorbing quasiprimary ideal of R /ψ ( R ) . Also, as I is not a 2-absorbing quasi primary idealof R , then I /ψ ( I ) is not a 2-absorbing quasi primary ideal of R /ψ ( R ) , byProposition 3(ii). Thus, I × R /φ ( I × R ) is a weakly 2-absorbing quasi primaryideal of R/φ ( I × R ) that is not a 2-absorbing quasi primary. Consequently, againby Proposition 2(ii) and Proposition 3(ii), we obtain that I × R is a φ -2-absorbingquasi primary ideal of R that is not a 2-absorbing quasi primary ideal of R. (cid:3) The following Theorem is a consequence of Theorem 11.
Theorem 13.
Let R and R be commutative rings with identity and let R = R × R . Then the following assertions are equivalent:(i) I × R is a weakly 2-absorbing quasi primary ideal of R. (ii) I × R is a 2-absorbing quasi primary ideal of R. (iii) I is a 2-absorbing quasi primary ideal of R . Theorem 14.
Let R and R be commutative rings with identity and R = R × R . Let I × I be a proper ideal of R, where I , I are nonzero ideals of R and R , respectively. Then the following assertions are equivalent:(i) I × I is a weakly 2-absorbing quasi primary ideal of R. (ii) I × I is a 2-absorbing quasi primary ideal of R. (ii) I = R and I is a 2-absorbing quasi primary ideal of R or I = R and I is a 2-absorbing quasi primary ideal of R or I , I are quasi primary of R , R , respectively.Proof. ( i ) ⇒ ( iii ) : Suppose that I × I is a weakly 2-absorbing quasi primary idealof R. If I = R , by Theorem 13, I is a 2-absorbing quasi primary ideal of R . Similarly, if I = R , I is a 2-absorbing quasi primary ideal of R . Thus we mayassume that I (cid:54) = R and I (cid:54) = R . Let us show I is a quasi primary ideal of R .Take x, y ∈ R such that xy ∈ I . Choose 0 (cid:54) = a ∈ I . Then 0 (cid:54) = ( a, , x )(1 , y ) =( a, xy ) ∈ I × I . By our hypothesis, ( a, x ) ∈ √ I × I = √ I × √ I or (1 , xy ) ∈√ I × √ I or ( a, y ) ∈ √ I × √ I . If (1 , xy ) ∈ √ I × √ I , a contradiction (as I (cid:54) = R ). Thus we obtain that ( a, x ) ∈ √ I × √ I or ( a, y ) ∈ √ I × √ I . Thisimplies that x ∈ √ I or y ∈ √ I . Similarly, we can show that I is a quasi primaryideal of R . ( ii ) ⇔ ( iii ) : By [21, Theorem 2.23].( ii ) ⇒ ( i ) : It is clear. (cid:3) Theorem 15.
Let R and R be commutative rings with identity and R = R × R . Then a non-zero ideal I × I of R is weakly 2-absorbing quasi primary that is not2-absorbing quasi primary if and only if one of the following assertions holds:(i) I (cid:54) = R is a nonzero weakly quasi primary ideal of R that is not quasiprimary and I = 0 is a quasi primary ideal of R . ENERALIZATION OF 2-ABSORBING QUASI PRIMARY IDEALS (ii) I (cid:54) = R is a nonzero weakly quasi primary ideal of R that is not quasiprimary and I = 0 is a quasi primary ideal of R . Proof.
Assume that I × I is a weakly 2-absorbing quasi primary ideal of R thatis not 2-absorbing quasi primary. Suppose that I (cid:54) = 0 and I (cid:54) = 0 . By Theorem14, I × I is 2-absorbing quasi primary, a contradiction. Thus I = 0 or I = 0 . Without loss of generality, suppose that I = 0 . Let us prove that I = 0 is aquasi primary ideal of R . Choose x, y ∈ R such that xy ∈ I . Take 0 (cid:54) = a ∈ I . Then 0 (cid:54) = ( a, , x )(1 , y ) = ( a, xy ) ∈ I × I . By our hypothesis, ( a, x ) ∈√ I × I = √ I × √ I or (1 , xy ) ∈ √ I × √ I or ( a, y ) ∈ √ I × √ I . Here (1 , xy ) / ∈√ I × √ I . Indeed, firstly observe that I (cid:54) = R . If I = R , then by Theorem11, I × I = R × a, x ) ∈ √ I × I = √ I × √ I or ( a, y ) ∈ √ I × √ I . This implies x ∈ √ I or y ∈ √ I . Hence I = 0 is quasi primary. Now, let us show that I is weaklyquasi primary ideal of R . Choose x, y ∈ R such that 0 (cid:54) = xy ∈ I . Then 0 (cid:54) = ( x, y, ,
0) = ( xy, ∈ I × I × I . As I × I is weakly2-absorbing quasi primary and ( xy, / ∈ √ I × , we have ( y, ∈ √ I × x, ∈ √ I × . This implies that x ∈ √ I or y ∈ √ I . Finally, we show that I is not quasi primary. Suppose that I is quasi primary. As I = 0 is a quasiprimary, we have that I × I is 2-absorbing quasi primary by [21, Theorem 2.3].This contradicts with our assumption. Thus I is not quasi primary. Conversely,assume that (i) holds. Let us prove I × I is weakly 2-absorbing quasi primary.Let (0 , (cid:54) = ( a , a )( b , b )( c , c ) ∈ I = I × I = I × . As a b c = 0 , we get a b c (cid:54) = 0 . Since a b c ∈ I and I is a quasi primary ideal of R , we get either a ∈ √ I or b ∈ √ I or c ∈ √ I . Without loss of generality, we may assumethat a ∈ √ I . On the other hand, since 0 (cid:54) = a b c = b ( a c ) ∈ I and I is aweakly quasi primary ideal, we have either b ∈ √ I or a c ∈ √ I . This impliesthat either ( a , a )( b , b ) ∈ √ I × I or ( a , a )( c , c ) ∈ √ I × I . In other cases,one can similarly show that ( a , a )( b , b ) ∈ √ I × I or ( a , a )( c , c ) ∈ √ I × I or ( b , b )( c , c ) ∈ √ I × I . Hence, I × I is weakly 2-absorbing quasi primaryideal of R. Also, since I is not quasi primary ideal, I × I is not a 2-absorbingquasi primary ideal by [21, Theorem 2.3]. (cid:3) Theorem 16.
Let R and R be commutative rings with identity and let R = R × R . Suppose that ψ i : L ( R i ) → L ( R i ) ∪ {∅} ( i = 1 , are functions and φ = ψ × ψ . Let I = I × I be a nonzero ideal of R and φ ( I ) (cid:54) = I × I . Then I × I is φ -2-absorbing quasi primary that is not 2-absorbing quasi primary if andonly if φ ( I ) (cid:54) = ∅ and one of the following statements holds.(i) ψ ( R ) = R and I is a ψ -2-absorbing quasi primary ideal of R that is nota 2-absorbing quasi primary ideal of R . (ii) ψ ( R ) = R and I is a ψ -2-absorbing quasi primary ideal of R that isnot a 2-absorbing quasi primary ideal of R . (iii) I = ψ ( I ) is a quasi primary of R and I (cid:54) = R is a ψ -quasi primaryideal of R that is not quasi primary such that I (cid:54) = ψ ( I ) (note that if I = 0 , then I (cid:54) = 0) (iv) I = ψ ( I ) is a quasi primary of R and I (cid:54) = R is a ψ -quasi primaryideal of R that is not quasi primary such that I (cid:54) = ψ ( I ) (note that if I = 0 , then I (cid:54) = 0) Proof.
Suppose that I × I is a φ -2-absorbing quasi primary ideal that is not2-absorbing quasi primary. Then φ ( I ) (cid:54) = ∅ . Let I = R . Then ψ ( R ) = R and I is a ψ -2-absorbing quasi primary ideal of R that is not a 2-absorbingquasi primary ideal of R by Theorem 12. Let I = R . Then ψ ( R ) = R and I is a ψ -2-absorbing quasi primary ideal of R that is not a 2-absorbing quasiprimary ideal of R by Theorem 12. Hence assume that I (cid:54) = R and I (cid:54) = R . Since φ ( I ) (cid:54) = I × I , we obtain that I/φ ( I ) is a nonzero weakly 2-absorbing quasiprimary ideal of R/φ ( I ) that is not a 2-absorbing quasi primary by Proposition2(ii). Thus I /ψ ( I ) × I /ψ ( I ) is a nonzero weakly 2-absorbing quasi primaryideal of R /ψ ( I ) × R /ψ ( I ) that is not a 2-absorbing quasi primary. Then byTheorem 15, we know that one of the following cases holds:Case 1: I /ψ ( I ) = ψ ( I ) /ψ ( I ) is a quasi primary ideal of R /ψ ( I ) and I /ψ ( I ) is a non-zero weakly quasi primary ideal of R /ψ ( I ) that is not quasiprimary.Case 2: I /ψ ( I ) = ψ ( I ) /ψ ( I ) is a quasi primary ideal of R /ψ ( I ) and I /ψ ( I ) is a non-zero weakly quasi primary ideal of R /ψ ( I ) that is not quasiprimary.Thus, (iii) or (iv) holds by Proposition 2(i) and Proposition 3(i).Conversely, assume that φ ( I ) (cid:54) = ∅ . If (i) or (ii) holds, then I × I is φ -2-absorbingquasi primary that is not 2-absorbing quasi primary by Theorem 12. Assume that(iii) or (iv) holds, then I/φ ( I ) is a non-zero weakly 2-absorbing quasi primaryideal of R/φ ( I ) that is not a 2-absorbing quasi primary by Theorem 15. Thus I × I is φ -2-absorbing quasi primary that is not 2-absorbing quasi primary of R by Proposition 2(ii) and Proposition 3(ii). (cid:3) Theorem 17.
Let R and R be commutative rings with identity and I , I benonzero ideals of R and R , respectively. Let R = R × R and ψ i : L ( R i ) → L ( R i ) ∪ {∅} ( i = 1 , be functions such that ψ ( I ) (cid:54) = I and ψ ( I ) (cid:54) = I . Supposethat φ = ψ × ψ and I × I is a proper ideal of R. Then the following assertionsare equivalent:(i) I × I is a φ -2-absorbing quasi primary ideal of R. (ii) Either I = R and I is a 2-absorbing quasi primary ideal of R or I = R and I is a 2-absorbing quasi primary ideal of R or I , I are quasi primary idealsof R and R , respectively.(iii) I × I is a 2-absorbing quasi primary ideal of R. Proof.
Assume that ψ ( I ) = ∅ or ψ ( I ) = ∅ . Then clearly φ ( I × I ) = ∅ so that( i ) ⇔ ( ii ) ⇔ ( iii ) follows from [21, Theorem 2.23]. Hence suppose that ψ ( I ) (cid:54) = ∅ and ψ ( I ) (cid:54) = ∅ , so φ ( I × I ) (cid:54) = ∅ . ( i ) ⇒ ( ii ) : Let I × I be a φ -2-absorbing quasi primary ideal of R. Thus I /ψ ( I ) × I /ψ ( I ) is a non-zero weakly 2-absorbing quasi primary ideal of R /ψ ( I ) × R /ψ ( I ) by Proposition 2(ii). Then by Theorem 14, we know thatone of the following cases holds: Case 1 : I /ψ ( I ) = R /ψ ( I ) and I /ψ ( I ) is a 2-absorbing quasi primaryideal of R /ψ ( I ) . Then we have I = R and I is a 2-absorbing quasi primaryideal of R . Case 2 : I /ψ ( I ) = R /ψ ( I ) and I /ψ ( I ) is a 2-absorbing quasi primaryideal of R /ψ ( I ). Similar to Case 1, I = R and I is a 2-absorbing quasi primaryideal of R . ENERALIZATION OF 2-ABSORBING QUASI PRIMARY IDEALS Case 3 : I /ψ ( I ) and I /ψ ( I ) are quasi primary of R /ψ ( I ) , R /ψ ( I ) , respectively. Then I , I are quasi primary ideals of R and R , respectively byProposition 3(ii).( ii ) ⇒ ( iii ) : Assume that I = R and I is a 2-absorbing quasi primary idealof R or I = R and I is a 2-absorbing quasi primary ideal of R or I , I arequasi primary ideals of R and R , respectively. Then by Theorem Theorem [21,Theorem 2.23], I × I is a 2-absorbing quasi primary ideal of R. ( iii ) ⇒ ( i ) : It is evident. (cid:3) References [1] Anderson, D. D., Batanieh, M., Generalizations of prime ideals,
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