On weakly 1-absorbing prime ideals of commutative rings
aa r X i v : . [ m a t h . A C ] F e b On weakly -absorbing prime ideals of commutative rings ∗† M. J. Nikmehr a , R. Nikandish b and A. Yassine a a Faculty of Mathematics, K.N. Toosi University of Technology,
P.O. BOX 16315-1618, Tehran, Iran nikmehr @ kntu . ac . ir yassine ali @ email . kntu . ac . ir b Department of Mathematics, Jundi-Shapur University of Technology,
P.O. BOX 64615-334, Dezful, Iran r . nikandish @ ipm . ir Abstract
Let R be a commutative ring with identity. In this paper, we introduce the conceptof weakly 1-absorbing prime ideals which is a generalization of weakly prime ideals.A proper ideal I of R is called weakly 1-absorbing prime if for all nonunit elements a, b, c ∈ R such that 0 = abc ∈ I , then either ab ∈ I or c ∈ I . A number of resultsconcerning weakly 1-absorbing prime ideals and examples of weakly 1-absorbing primeideals are given. It is proved that if I is a weakly 1-absorbing prime ideal of a ring R and 0 = I I I ⊆ I for some ideals I , I , I of R such that I is free triple-zerowith respect to I I I , then I I ⊆ I or I ⊆ I . Among other things, it is shownthat if I is a weakly 1-absorbing prime ideal of R that is not 1-absorbing prime, then I = 0. Moreover, weakly 1-absorbing prime ideals of PID’s and Dedekind domainsare characterized. Finally, we investigate commutative rings with the property thatall proper ideals are weakly 1-absorbing primes. We assume throughout this paper that all rings are commutative with identity. Let R be a ring and I be an ideal of R . The set of nilpotent elements of R , the set of zero-divisors of R , the set of integers, and integers modulo n are denoted by √ Z ( R ), Z and ∗ Key Words : 1-absorbing prime ideal, Weakly 1-absorbing prime ideal, Prime ideal, Weakly prime ideal. † Mathematics Subject Classification : 13A15, 13C05. n , respectively. By a proper ideal I of R we mean an ideal with I = R . A ring R iscalled local if it has a unique maximal ideal. A ring R is called a reduced ring if it has nonon-zero nilpotent elements; i.e., √ P of R such that for a, b ∈ K with ab ∈ P , either a ∈ P or b ∈ P where K is the quotient field of R . In 2003, Anderson and Smith [1] introducedthe notion of a weakly prime ideal, i.e., a proper ideal P of R with the property that for a, b ∈ R , 0 = ab ∈ P implies a ∈ P or b ∈ P . So a prime ideal is weakly prime. In2005, Bhatwadekar and Sharma [6] introduced the notion of almost prime ideal which isalso a generalization of prime ideal. A proper ideal I of an integral domain R is said tobe almost prime if for a, b ∈ R with ab ∈ I \ I , then either a ∈ I or b ∈ I , and it isclear that every weakly prime ideal is an almost prime ideal. Another generalization ofprime ideal is 2-prime; Indeed, a nonzero proper ideal I of R is called 2-prime if whenever a, b ∈ R and ab ∈ I , then a ∈ I or b ∈ I (See [5] and [9] for more details). The notionof 2-absorbing ideals was introduced and investigated in 2007 by Badawi [2]. A nonzeroproper ideal I of R is called 2-absorbing if whenever a, b, c ∈ R and abc ∈ I , then ab ∈ I or ac ∈ I or bc ∈ I . In [3], Badawi and Darani extended the concept of weakly primeideal to weakly 2-absorbing ideal. A proper ideal I of R is said to be a weakly 2-absorbingideal of R if whenever a, b, c ∈ R with 0 = abc ∈ I implies ab ∈ I or ac ∈ I or bc ∈ I . In[4], 1-absorbing primary ideal was introduced and studied. A proper ideal I of R is called1-absorbing primary if for all nonunit a, b, c ∈ R such that abc ∈ I , then either ab ∈ I or c ∈ √ I . Recall that 1-absorbing prime ideals, as a generalization of prime ideals, wereintroduced and investigated in [11]. In this paper, we extend the concepts of weakly primeideal and 1-absorbing primary ideal to weakly 1-absorbing prime ideal.A proper ideal I of R is called (weakly) -absorbing prime if for all nonunit elements a, b, c ∈ R such that (0 =) abc ∈ I , then either ab ∈ I or c ∈ I . Clearly, every weakly primeideal is a weakly 1-absorbing prime ideal. However, the converse is not true.This paper is organized as follows. In section 2, first we give the definition of weakly1-absorbing prime ideals (Definition 2.1). For nontrivial weakly 1-absorbing prime idealssee Example 2.3. Also, it is proved (Theorem 2.6) that if I is a weakly 1-absorbing primeideal of R that is not 1-absorbing prime, then I = 0. In Theorem 2.10 we give a conditionunder which every weakly 1-absorbing prime ideal of R is 1-absorbing prime. Among otherthings, it is shown (Theorem 2.12) that the radical of a weakly 1-absorbing prime ideal ofa ring R need not be a prime ideal of R . Finally, it is proved (Theorem 2.23) that if I is aweakly 1-absorbing prime ideal of a ring R and 0 = I I I ⊆ I for some ideals I , I , I of2 such that I is free triple-zero with respect to I I I , then I I ⊆ I or I ⊆ I . In Section3, we characterize rings with the property that all proper ideals are weakly 1-absorbingprime. -absorbing prime ideals In this section, the concept of weakly 1-absorbing prime ideals is introduced and in-vestigated. We start with the following definitions.
Definition 2.1
Let R be a ring. A proper ideal I of R is called weakly -absorbingprime if for all nonunit elements a, b, c ∈ R such that = abc ∈ I , then ab ∈ I or c ∈ I . Definition 2.2
Suppose that I is a weakly -absorbing prime ideal of a ring R and a, b, c ∈ R are nonunit elements. (1) We say ( a, b, c ) is a triple-zero of I if abc = 0 , ab / ∈ I and c / ∈ I . (2) Suppose that I I I ⊆ I , for some ideals I , I , I of R . We say I is free triple-zerowith respect to I I I , if ( a, b, c ) is not a triple-zero of I , for every a ∈ I , b ∈ I , and c ∈ I . It is easy to see that if I is a weakly 1-absorbing prime ideal that is not 1-absorbingprime, then there exists a triple-zero of I . Note that every weakly prime ideal is a weakly1-absorbing prime ideal, and every weakly 1-absorbing prime ideal is a weakly 2-absorbingprime ideal. The following example provides a weakly 1-absorbing prime ideal that is not1-absorbing prime, a weakly 2-absorbing ideal that is not weakly 1-absorbing prime anda weakly 1-absorbing prime ideal that is not weakly prime. Example 2.3 (1) Consider the ideal J = { , } of Z . Let R = Z (+) J and I = { (0 , , (0 , } . It is easy to see that I is a weakly 1-absorbing prime ideal, since abc ∈ I for some a, b, c ∈ R \ I if and only if abc = (0 , , , , ∈ I and both(4 , / ∈ I and (2 , / ∈ I , thus I is a weakly 1-absorbing prime ideal that is not 1-absorbingprime.(2) Suppose that R = R × R × R , where R , R , R are fields and I = R ×{ } × { } . One can easily see that the ideal I is not weakly 1-absorbing prime, since30 , , = (1 , , , , , ,
0) = (1 , , ∈ I and neither (1 , , , ,
1) = (1 , , ∈ I nor (1 , , ∈ I . But I is a weakly 2-absorbing ideal of R , by [3, Theorem 3.5]. Henceevery weakly 2-absorbing ideal need not be weakly 1-absorbing prime.(3) Consider the ideal J = { , , } of the ring Z . Clearly, J is a weakly 1-absorbingprime ideal of Z that is not weakly prime. Proposition 2.4
Let R be a ring, I a weakly -absorbing prime ideal of R and c be anonunit element of R \ I . Then ( I : c ) is a weakly prime ideal of R . Proof.
Assume that 0 = ab ∈ ( I : c ) for some nonunit element c ∈ R \ I such that a / ∈ ( I : c ). We may assume that a, b are nonunit elements of R . As 0 = abc = acb ∈ I and ac / ∈ I and I is a weakly 1-absorbing prime ideal of R , we have b ∈ I ⊆ ( I : c ). Hence( I : c ) is a weakly prime ideal of R . (cid:3) Let R be a local ring and I be a weakly 1-absorbing prime ideal of R which is not 1-absorbing prime. To prove I = 0, the following lemma is needed. Lemma 2.5
Let I be a weakly -absorbing prime ideal of a local ring R and let ( a, b, c ) be a triple-zero of I for some nonunit elements a, b, c ∈ R . Then the following statementshold. (1) abI = acI = bcI = 0 . (2) aI = bI = cI = 0 . Proof. (1) Assume that abI = 0. Then there exists x ∈ I such that abx = 0.Therefore ab ( c + x ) = 0. Since ab / ∈ I and I is a weakly 1-absorbing prime ideal, ( c + x ) ∈ I ,and hence c ∈ I , a contradiction. Thus abI = 0. Now suppose that acI = 0. Then thereexists x ∈ I such that acx = 0. Therefore a ( b + x ) c = 0. Since c / ∈ I and I is a weakly1-absorbing prime ideal, a ( b + x ) ∈ I , and so ab ∈ I , a contradiction. Thus acI = 0.Similarly, one can easily see that bcI = 0.(2) Assume that axy = 0 for some x, y ∈ I . By part (1), a ( b + x )( c + y ) = axy = 0.Since R is local, the set of nonunit elements of R is an ideal of R . Therefore ( b + x ) , ( c + y )are nonunit elements of R . This means that either a ( b + x ) ∈ I or c + y ∈ I . Henceeither ab ∈ I or c ∈ I , a contradiction. Thus aI = 0. Similarly, one can easily show that bI = cI = 0. (cid:3) Theorem 2.6
Let I be a weakly -absorbing prime ideal of a local ring R that is not -absorbing prime. Then I = 0 . roof. Suppose that I is a weakly 1-absorbing prime ideal of a ring R that is not1-absorbing prime. Then there exists a triple-zero ( a, b, c ) of I for some nonunit elements a, b, c ∈ R . Suppose that I = 0. Hence xyz = 0 for some x, y, z ∈ I . By Lemma 2.5,( a + x )( b + y )( c + z ) = xyz = 0. Since R is local, the set of nonunit elements of R isan ideal of R . Therefore ( a + x ) , ( b + y ) and ( c + z ) are nonunit elements. Hence either( a + x )( b + y ) ∈ I or ( c + z ) ∈ I , and so either ab ∈ I or c ∈ I , a contradiction. Thus I = 0. (cid:3) It is worth mentioning that if I is an ideal of a ring R such that I = 0, then I neednot be a weakly 1-absorbing prime ideal. For example, let R = Z / Z and I = 8 Z / Z .Then I = 0, but 0 = 2 · · ∈ I and 2 , / ∈ I .Now, we state the following corollary. Corollary 2.7 (1)
Let I be a weakly -absorbing prime ideal of a local ring R that isnot -absorbing prime. Then √ I = √ . (2) If R is a reduced local ring and I = 0 is a proper ideal of R , then I is a weakly -absorbing prime ideal of R if and only if I is a -absorbing prime ideal of R . Theorem 2.8
Let I be a weakly -absorbing prime ideal of a local ring R that is not -absorbing prime. Then the following statements hold. (1) If w ∈ √ , then either w ∈ I or w I = wI = { } . (2) √ I = { } . Proof. (1) Suppose that w ∈ √ w I = { } . Suppose also that n is theleast positive integer such that w n = 0. Then n ≥ w ( x + w n − ) = w x = 0 forsome x ∈ I . Therefore, either w ∈ I or ( x + w n − ) ∈ I , since R is local (which means that( x + w n − ) is nonunit) and I is weakly 1-absorbing prime. If ( x + w n − ) ∈ I , then w n − ∈ I ,and so w ∈ I , as w n − / ∈ I . Hence w ∈ I . Thus either w ∈ I or w I = { } , for every w ∈ √
0. Next, we show that wI = { } . Suppose that wI = { } and w / ∈ I . Then w I = { } and wxy = 0 for some x, y ∈ I . Let m be the least positive integer such that w m = 0. This means that m ≥ w I = 0. Therefore w ( w + x )( w m − + y ) = wxy = 0.But R is local (which means that ( y + w n − ) is non unit) and I is weakly 1-absorbingprime, hence either w ∈ I or w m − ∈ I , a contradiction. Thus wI = { } .(2) Suppose that a, b ∈ √
0. It follows from part (1) that if either a / ∈ I or b / ∈ I , then abI = { } . Hence assume that a ∈ I and b ∈ I , which means that ab ( a + b ) ∈ I . Wedistinguish two cases, according as ( a, b, a + b ) is a triple-zero of I or not. If ( a, b, a + b ) is5 triple-zero of I , then abI = { } , by Lemma 2.5 (1), and thus √ I = { } . If ( a, b, a + b )is not a triple-zero of I , then it is easily seen that ab ∈ I , and so √ I = { } , by Theorem2.6. (cid:3) Corollary 2.7 and Theorem 2.8 lead to the following corollary.
Corollary 2.9
Let
I, J, K be weakly -absorbing prime ideals of a local ring R suchthat none of them is -absorbing prime. Then I J K = IJ K = IJ K = I J = I K = J K = { } . If I is a proper ideal of R , it is well known that √ I is a prime ideal of R if and only if √ I is a primary ideal of R . In the following result, we give a condition under which everyweakly 1-absorbing prime ideal of R is 1-absorbing prime. Theorem 2.10
Suppose that I is a proper ideal of R such that √ ⊆ I and √ is aprime (primary) ideal of R . Then I is a weakly -absorbing prime ideal of R if and onlyif I is a -absorbing prime ideal of R . Proof.
Suppose that I is a weakly 1-absorbing prime ideal of the ring R and xyz ∈ I for some nonunit elements x, y, z ∈ R . If xyz = 0, then either xy ∈ I or z ∈ I . Thereforeassume that xyz = 0 and z / ∈ I . Since xyz = 0 ∈ √ R , weconclude that xy ∈ √ ⊆ I . Thus I is a weakly 1-absorbing prime ideal of R if and onlyif I is a 1-absorbing prime ideal of R . (cid:3) In the following result, we give a condition under which a weakly 1-absorbing primeideal of R is not 1-absorbing prime. Theorem 2.11
Suppose that I is a weakly -absorbing prime ideal of a local ring R and { } has a triple-zero ( a, b, c ) for some nonunit elements a, b, c ∈ R such that ab / ∈ √ .Then I is not a -absorbing prime ideal of R if and only if I ⊆ √ . Proof.
Suppose that I is not a 1-absorbing prime ideal of R . Hence, by part (1) ofCorollary 2.7, I ⊆ √
0. Conversely, suppose that I ⊆ √ { } has a triple-zero ( a, b, c )for some nonunit elements a, b, c ∈ R such that ab / ∈ √
0. Then ab / ∈ I and c / ∈ I . Hence( a, b, c ) is a triple-zero of I , and thus I is not a 1-absorbing prime ideal of R . (cid:3) One can easily see that if I is a 1-absorbing prime ideal of a ring R , then there existsjust one prime ideal of R that is minimal over I . In the following result, we show that for6very positive integer n ≥
2, there exist a ring R and a nonzero weakly 1-absorbing primeideal I of R such that there are exactly n prime ideals of R that are minimal over I . Theorem 2.12
Suppose that n ≥ is a positive integer. Then there exist a ring R and a nonzero weakly -absorbing prime ideal I of R such that I has exactly n minimalprime ideals of R . Proof.
Suppose that n ≥ D = Z × · · · × Z ( n times).Clearly, M = { , } is an ideal of Z . Define the D -module M such that xM = a M forevery x = ( a , . . . , a n ) ∈ D and consider the idealization ring R = D (+) M and the ideal I = { (0 , . . . , } (+) M of R . It follows from [8, Theorem 25.1 (3)] that every prime idealof R is of the form P (+) M for some prime ideal P of D and since for every a, b, c ∈ R \ I and abc ∈ I , we deduce that abc = ((0 , . . . , , I is a nonzero weakly 1-absorbingprime ideal of R , and thus there are exactly n prime ideals of R that are minimal over I . (cid:3) In the next result, one can see that in a reduced ring, the radical of a nonzero weakly1-absorbing prime ideal a prime ideal.
Theorem 2.13
Let R be a reduced ring and I a nonzero weakly -absorbing primeideal of R . Then √ I is a prime ideal of R . Proof.
Let 0 = ab ∈ √ I for some a, b ∈ R . Without loss of generality, we may assumethat a, b are nonunit. Then ( ab ) n ∈ I for some positive integer n , and so ( ab ) n = 0, because R is reduced. Therefore 0 = a m a ( n − m ) b n ∈ I for some positive integer m , and hence either a m a ( n − m ) = a n ∈ I or b n ∈ I . Thus √ I is a weakly prime ideal of R. But R is reducedand I = { } , hence √ I is a prime ideal of R , by [1, Corollary 2]. (cid:3) It was shown in [11, Theorem 2.4] that if R is a non-local ring, then every 1-absorbingprime ideal is prime. In the following theorem, we show that if R is a non-local ring and I is a proper ideal of R having the property that ann ( x ) is not a maximal ideal of R forevery element x ∈ I , then I is a weakly 1-absorbing prime ideal if and only if I is a weaklyprime ideal. Theorem 2.14
Let R be a non-local ring and I a proper ideal of R having the propertythat ann ( x ) is not a maximal ideal of R for every element x ∈ I . Then I is a weakly -absorbing prime ideal if and only if I is a weakly prime ideal. roof. It is easy to see that every weakly prime ideal of R is a weakly 1-absorbingprime ideal of R . Then assume that I is a weakly 1-absorbing prime ideal of R and let0 = ab ∈ I for some a, b ∈ R . Without loss of generality, we may assume that a, b arenonunit. Since ab = 0, ann ( ab ) is a proper ideal of R , and so ann ( ab ) ⊂ L , for somemaximal ideal L of R . But R is a non-local ring, hence there exists a maximal ideal M of R such that M = L . Suppose that m ∈ M \ L . Then m / ∈ ann ( ab ), which means that0 = mab ∈ I . Since I is a weakly 1-absorbing prime ideal of R , either ma ∈ I or b ∈ I . If b ∈ I , then the proof is complete. So suppose that b / ∈ I . Therefore ma ∈ I . But m / ∈ L and L is a maximal ideal of R , hence m / ∈ J( R ). This means that there exists an r ∈ R such that 1 + rm is a nonunit element of R . We take the following two cases: Case one: if1 + rm / ∈ ann ( ab ), then 0 = (1 + rm ) ab / ∈ I . Since I is a weakly 1-absorbing prime ideal of R and b / ∈ I , we conclude that (1 + rm ) a = a + rma ∈ I . But rma ∈ I , so a ∈ I and I is aweakly prime ideal of R . Case two:
Suppose that 1 + rm ∈ ann ( ab ). Since ann ( ab ) is nota maximal ideal of R and ann ( ab ) ⊂ L , there exists an element w ∈ L \ ann ( ab ). Therefore0 = wab ∈ I . But I is a weakly 1-absorbing prime ideal of R and b / ∈ I , thus wa ∈ I . Wehave 1 + rm + w is a nonzero nonunit element of L , because 1 + rm ∈ ann ( ab ) ⊂ L and w ∈ L \ ann ( ab ). Thus 0 = (1 + rm + w ) ab ∈ I . Now, since I is a weakly 1-absorbingprime ideal of R and b / ∈ I , we have (1 + rm + w ) a = a + rma + wa ∈ I . Hence a ∈ I ,and thus I is a weakly prime ideal of R . (cid:3) Let R be an integral domain with the quotient field K . Recall that a proper ideal I of R is called invertible if II − = R , where I − = { r ∈ K : rI ⊆ R } . An integral domain iscalled a Dedekind domain if every nonzero proper ideal of R is invertible. In the followingresults, weakly 1-absorbing prime ideals of Dedekind domains and principal ideal domainsare completely described. Theorem 2.15
Let R be a Noetherian integral domain that is not a field and I anideal of R . Then (1) ⇒ (2) ⇒ (3) . (1) R is a Dedekind domain; (2) If I is a weakly -absorbing prime ideal of R , then I = M or I = M where M isa maximal ideal of R ; (3) If I is a weakly -absorbing prime ideal of R , then I = P or I = P where P = √ I is a prime ideal of R . Proof. (1) ⇒ (2) Suppose that R is a Noetherian integral domain that is not afield and I is a weakly 1-absorbing prime ideal of R such that P = √ I . Since R is aDedekind domain, we conclude that every nonzero prime ideal of R is a maximal ideal of8 . Therefore P is a maximal ideal of R , by Theorem 2.13. This means that I is a primaryideal of R such that P ⊆ I . Hence, by [11, Theorem 2.10], I = M or I = M where M is a maximal ideal of R .(2) ⇒ (3) This is obvious. (cid:3) In view of Theorem 2.15, we have the following result.
Corollary 2.16
Let R be a principal ideal domain that is not a field and I be a nonzeroproper ideal of R . If I is a weakly -absorbing prime ideal of R , then I = pR or I = p R for some nonzero prime element p of R . In the following theorems we show that weakly 1-absorbing prime ideals are really ofinterest in indecomposable rings.
Theorem 2.17
Suppose that R = R × R is a decomposable ring, where R and R are rings and I is a proper ideal of R . Then the following statements are equivalent. (1) I × R is a weakly -absorbing prime ideal of R . (2) I × R is a -absorbing prime ideal of R . (3) I is a -absorbing prime ideal of R . Proof. (3) ⇒ (2) and (2) ⇒ (1) These are clear.(1) ⇒ (3) Suppose that I × R is a weakly 1-absorbing prime ideal of R and abc ∈ I for some nonunit elements a, b, c ∈ R . Suppose also that 0 = x ∈ R . Then (0 , =( a, b, c, x ) = ( abc, x ) ∈ I × R , and so either ( a, b,
1) = ( ab, ∈ I × R or( c, x ) ∈ I × R . Hence, either ab ∈ I or c ∈ I . Thus I is a 1-absorbing prime ideal of R . (cid:3) Theorem 2.18
Suppose that R = R × R is a decomposable ring, where R and R are rings and I , I are nonzero ideals of R and R , respectively. Then the followingstatements are equivalent. (1) I × I is a weakly -absorbing prime ideal of R . (2) I = R and I is a -absorbing prime ideal of R or I = R and I is a -absorbingprime ideal of R or I , I are prime ideals of R , R , respectively. (3) I × I is a -absorbing prime ideal of R . (4) I × I is a prime ideal of R . roof. (1) ⇒ (2) Let I × I be a weakly 1-absorbing prime ideal of R . If I = R ( I = R ), then by Theorem 2.17, I ( I ) is a 1-absorbing prime ideal of R ( R ). Thereforeassume that I and I are proper ideals. Suppose that a, b ∈ R such that ab ∈ I and0 = x ∈ I . Then (0 , = ( x, , a )(1 , b ) = ( x, ab ) ∈ I × I . But I is proper, hence(1 , a ) / ∈ I × I and (1 , b ) / ∈ I × I . Without loss of generality, we may assume that x, a, b are nonunit. Since I × I is a weakly 1-absorbing prime ideal of R , we conclude that( x, , a ) = ( x, a ) ∈ I × I . Thus I is a prime ideal of R . Similarly, it can be easilyshown that I is a prime ideal of R .(2) ⇒ (3) If I = R and I is a 1-absorbing prime ideal of R or I = R and I is a1-absorbing prime ideal of R , then the result follows from Theorem 2.17. Assume that I , I are prime ideals of R , R , respectively. Since I , I are nonzero ideals of R and R ,assume that (0 , = ( a, b ) ∈ I × I . Then (0 , = ( a, a, , b ) = ( a , b ) ∈ I × I .This means that either ( a, a, ∈ I × I or (1 , b ) ∈ I × I , since I × I is a weakly1-absorbing prime ideal of R . Hence either ( a, ∈ I × I or (1 , b ) ∈ I × I , as I is aprime ideals of R , a contradiction. Thus I × I = R × I or I × I = I × R and inboth cases I × I is a 1-absorbing prime ideal of R .(3) ⇒ (4) Since R is not a local ring, the proof follows from [11, Theorem 2.4].(4) ⇒ (1) This is straightforward. (cid:3) Theorem 2.19
Suppose that R = R × R is a decomposable ring, where R and R are rings and I is a nonzero proper ideal of R and I is a proper ideal of R . Then (1) ⇒ (2) . In fact, (2) ⇒ (1) need not be true. (1) I × I is a weakly -absorbing prime ideal of R that is not a -absorbing primeideal. (2) I is a weakly prime ideal of R that is not a prime ideal and I = { } is a primeideal of R . Proof. (1) ⇒ (2) Suppose that I × I is a weakly 1-absorbing prime ideal of R thatis not a 1-absorbing prime ideal and I = { } . Hence, by Theorem 2.18, I × I is a1-absorbing prime ideal of R , a contradiction, and thus I = { } . Now, we show that I is a prime ideal of R . Suppose that a, b ∈ R such that ab ∈ I and 0 = x ∈ I . Then(0 , = ( x, , a )(1 , b ) = ( x, ab ) ∈ I × I . But I is proper, hence (1 , a ) / ∈ I × I and(1 , b ) / ∈ I × I . Without loss of generality, we may assume that x, a, b are nonunit. Since I × I is a weakly 1-absorbing prime ideal of R , ( x, , a ) = ( x, a ) ∈ I × I . Thus I = { } is a prime ideal of R . We show that I is a weakly prime ideal of R . Let a, b ∈ R suchthat 0 = ab ∈ I . Without loss of generality, we may assume that a, b are nonunit. Since100 , = ( b, , ab,
1) = ( ab , ∈ I × I and ( ab, / ∈ I × { } and I × I is a weakly1-absorbing prime ideal of R , we conclude that ( b, ∈ I × { } , and hence b ∈ I . Thus I is a weakly prime ideal of R . Assume that I is a prime ideal of R . Since I is a nonzeroideal of R , assume that 0 = a ∈ I . Then (0 , = ( a, a, ,
0) = ( a , ∈ I × I .This means that either ( a, a, ∈ I × I or (1 , ∈ I × I , as I × I is a weakly1-absorbing prime ideal of R . Hence either ( a, ∈ I × I or (1 , ∈ I × I , as I is aprime ideal of R , a contradiction. Thus I is a weakly prime ideal of R that is not aprime ideal and I = { } is a prime ideal of R .(2) ⇒ (1) Assume that I is a weakly prime ideal of R that is not a prime ideal and I = { } is a prime ideal of R . We show that I × I need not be a weakly 1-absorbingprime ideal of R . Since I = 0, there exists 0 = x ∈ I , and so (0 , = ( a, a, ,
0) =( a , ∈ I × { } . Since neither ( a, a,
1) = ( a , ∈ I × { } nor (1 , / ∈ I × { } and I is proper, we conclude that I × I is not a weakly 1-absorbing prime ideal of R . (cid:3) In the following, we show that in a decomposable ring R = R × R × R every nonzeroweakly 1-absorbing prime ideal of R is 1-absorbing prime. Theorem 2.20
Suppose that R = R × R × R is a decomposable ring, where R , R and R are rings and I = I × I × I is a nonzero proper ideal of R . Then I is a weakly -absorbing prime ideal if and only if I is a -absorbing prime ideal. Proof.
Let I = I × I × I be a nonzero weakly 1-absorbing prime ideal of R . Thenthere exists an element (0 , , = ( a, b, c ) ∈ I . Since ( a, , , b, , , c ) = ( a, b, c ),either ( a, b, ∈ I or (1 , , c ) ∈ I . Hence either I = R or I = R and I = R , and so I = I × I × R or I = R × R × I . Thus, by Theorem 2.17, I is a 1-absorbing primeideal of R . The converse is clear. (cid:3) Corollary 2.21
Suppose that R = R × R × R is a decomposable ring, where R , R and R are rings and I = I × I × I is a nonzero ideal of R . Then the followingstatements are equivalent. (1) I = I × I × I is a weakly -absorbing prime ideal of R . (2) I = I × I × I is a -absorbing prime ideal of R . (3) Either I = I × I × I such that for some k ∈ { , , } , I k is a -absorbing primeideal of R k , and I j = R j for every j ∈ { , , } − { k } , or I = I × I × I such that forsome k, m ∈ { , , } , I k is a prime ideal of R k , I m is a prime ideal of R m , and I j = R j for every j ∈ { , , } − { k, m } . I = I × I × I is a prime ideal of R . Proof. (1) ⇔ (2) It follows from Theorem 2.20.(2) ⇒ (3) Since I = I × I × I is a 1-absorbing prime ideal of R , we have either I = R or I = R or I = R . If I = R , then by the proof of Theorem 2.20, I = R and I = R , and so we are done. If I = R , then by the proof of Theorem 2.20, either I = R and I = R or I = R and I = R . The first case is clear, so assume that I = R and I = R . We show that I is a prime ideal of R and I is a prime of R . Suppose that a, b ∈ R such that ab ∈ I , and c, d ∈ R such that cd ∈ I . Then(0 , , = ( a, , , c, b, d,
1) = ( ab, cd, ∈ I . Hence either a ∈ I or b ∈ I , and thus I is a prime ideal of R . Similarly, since (0 , , = ( a, , b, c, , d,
1) = ( ab, cd, ∈ I ,we conclude that either c ∈ I or d ∈ I . Hence I is a prime ideal of R . Finally, assumethat I = R and I = R . By an argument similar to that we applied above, we concludethat I is a prime ideal of R and I is a prime ideal of R .(3) ⇒ (2) This is clear.(2) ⇔ (4) It follows from [11, Theorem 2.4]. (cid:3) The next theorem states that if I is a weakly 1-absorbing prime ideal of a ring R and0 = I I I ⊆ I for some ideals I , I , I of R such that I is free triple-zero with respect to I I I , then I I ⊆ I or I ⊆ I . First, we need the following lemma. Lemma 2.22
Let I be a weakly -absorbing prime ideal of a ring R . If abJ ⊆ I forsome nonunit elements a, b ∈ R and a proper ideal J of R such that ( a, b, c ) is not atriple-zero of I for every c ∈ J , then ab ∈ I or J ⊆ I . Proof.
Suppose that abJ ⊆ I , but ab / ∈ I and J * I . Then there exists an element j ∈ J \ I . But ( a, b, j ) is not a triple-zero of I and abj ∈ I and ab / ∈ I and j / ∈ I , acontradiction. (cid:3) Theorem 2.23
Suppose that I is a proper ideal of a ring R . Then the followingstatements are equivalent. (1) I is a weakly -absorbing prime ideal of R . (2) For any proper ideals I , I , I of R such that = I I I ⊆ I and I is free triple-zerowith respect to I I I , we have either I I ⊆ I or I ⊆ I . Proof. (1) ⇒ (2) Suppose that I is a weakly 1-absorbing prime ideal of R and0 = I I I ⊆ I for some proper ideals I , I , I of R such that I I * I and I is free12riple-zero with respect to I I I . Then there are nonunit elements a ∈ I and b ∈ I suchthat ab / ∈ I . Since abI ⊆ I , ab / ∈ I and ( a, b, c ) is not a triple-zero of I for every c ∈ J , itfollows from Lemma 2.22 that J ⊆ I .(2) ⇒ (1) Suppose that 0 = abc ∈ I for some nonunit elements a, b, c ∈ R and ab / ∈ I .Suppose also that I = aR, I = bR , and I = cR . Then 0 = I I I ⊆ I and I I * I .Hence I = cR ⊆ I , and thus c ∈ I . (cid:3) Suppose that I is an ideal of a ring R . It was shown in [11, Theorem 2.7], if I is 1-absorbing prime and I I I ⊆ I for some proper ideals I , I , I of R , then I I ⊆ I or I ⊆ I . We end this section with the following question: if I is weakly 1-absorbing primeand 0 = I I I ⊆ I for some proper ideals I , I , I of R , does it imply that I I ⊆ I or I ⊆ I ? -absorbingprime In this section we study rings in which every proper ideal is weakly 1-absorbing prime.To prove Theorem 3.2, the following lemma is needed.
Lemma 3.1
Let R be a ring. Then for every a, b, c ∈ J ( R ) , the ideal I = Rabc isweakly -absorbing prime if and only if abc = 0 . Proof.
Suppose that R is a ring, a, b, c ∈ J ( R ) and I = Rabc is an ideal of R . Onecan easily see that I is a weakly 1-absorbing prime ideal of R , if abc = 0. So assume that I is a weakly 1-absorbing prime ideal of R and abc = 0. Hence a, b, c are nonunit, and soeither ab ∈ I or c ∈ I . If ab ∈ I , then ab = abcx for some x ∈ R , and thus ab (1 − cx ) = 0.This follows that ab = 0, since cx ∈ J ( R ) which means that 1 − cx is unit. Therefore abc = 0, a contradiction. If c ∈ I , by a similar argument, we can see that c (1 − abk ) = 0for some x ∈ R . This implies that c = 0, a contradiction. Thus abc = 0. (cid:3) Theorem 3.2
Let R be a local ring with a unique maximal ideal M . Then everyproper ideal of R is weakly -absorbing prime if and only if M = { } . Proof.
Suppose that R is local with a maximal ideal M and every proper ideal of R is weakly 1-absorbing prime. It follows from Lemma 3.1 that abc = 0 for every a, b, c ∈ M ,since abcR is a weakly 1-absorbing prime ideal. Therefore M = { } . Conversely, supposethat M = { } and I is a nonzero proper ideal of R . Suppose also that 0 = abc ∈ I for13ome nonunit elements a, b, c of R . Then a, b, c ∈ M . But M = { } and 0 = abc , henceeither a is a unit of R or b is a unit of R or c is a unit of R , a contradiction. Thus I is aweakly 1-absorbing prime ideal. (cid:3) Corollary 3.3
Suppose that R is a local ring with a unique maximal ideal M suchthat M = { } . Then every proper ideal of R is a -absorbing prime ideal of R . Proof.
Suppose that R is local with a maximal ideal M and I is a proper ideal of R . It follows from Theorem 3.2 that I is weakly 1-absorbing prime, because M = { } .Therefore assume that 0 = abc ∈ I for some nonunit elements a, b, c of R . This meansthat a, b, c ∈ M . Since M = { } , abc = 0 and M is primary, we conclude that either c ∈ M ⊆ I or ab ∈ M . In the second case, it easy to see that ab = 0 ∈ I . Thus I is a1-absorbing prime ideal of R . (cid:3) Theorem 3.4
Suppose that R and R are rings and R = R × R . If R and R arelocal with maximal ideals M and M , respectively, and every proper ideal of R is a weakly -absorbing prime ideal of R , then M , M are zero ideals and either R or R is a field. Proof.
Suppose that R and R are local rings with maximal ideals M and M ,respectively, and R = R × R . First, suppose that every proper ideal of R is a weakly 1-absorbing prime ideal of R and a, b ∈ M such that ab = 0. Then the ideal I = abR ×{ } of R is a weakly 1-absorbing prime ideal of R . Without loss of generality, we may assume that a, b are nonunit. But (0 , = ( a, b, ,
0) = ( ab, ∈ I and I is a weakly 1-absorbingprime ideal of R , hence (1 , ∈ I , since ( a, b, / ∈ I , and so 1 = abx for some x ∈ R .This means that 1 − abx = 0, a contradiction, as 1 − abx is a unit element of R . Therefore M = { } . By a similar argument as above and taking the ideal I = { }× abR , one can seethat M = { } . Now, assume that R and R are not fields and look for a contradiction. Itfollows that M = { } , and so the ideal J = M ×{ } is a weakly 1-absorbing prime ideal of R . But M = { } and R is not a field, hence there exists c ∈ M such that c = 0. Supposethat 0 = m ∈ M . Then (0 , = ( m , , c )(1 , c ) = ( m , c ) = ( m , ∈ J = M × { } which is a contradiction, since both ( m , , c ) = ( m , c ) / ∈ J and (1 , c ) / ∈ J . Thus either R or R is a field. (cid:3) Finally, we characterize rings in which every ideal is weakly 1-absorbing prime. To thisend, the following lemma is needed.
Lemma 3.5 (1)
Suppose that R = R × R × R , where R , R and R are rings. Ifevery proper ideal of R is weakly -absorbing prime, then R , R , R are fields. Let R be a ring and every proper ideal of R be weakly -absorbing prime. Then R has at most three maximal ideals. Proof. (1) Let R = R × R × R be a decomposable ring, where R , R and R are rings such that every proper ideal of R is weakly 1-absorbing prime. Without lossof generality, assume that R is not a field. Let J be a non-zero proper ideal of R and m be a non-zero element of J . Suppose that I = J × { } × { } . Then I is a weakly1-absorbing prime ideal of R . Since (0 , , = ( m, , , , , ,
0) = ( m, , ∈ I andboth ( m, , , ,
1) = ( m, , / ∈ I and (1 , , / ∈ I , we have a contradiction. Hence R , R , R are fields.(2) Let M , M , M , M be distinct maximal ideals of R and I = M ∩ M ∩ M . Sinceevery weakly 1-absorbing prime is weakly 2-absorbing, and by [2, Theorem 2.5], I is not a2-absorbing ideal of R , we conclude that I is a weakly 2-absorbing ideal of R that is nota 2-absorbing ideal of R . Therefore, by [3, Theorem 2.4], I = { } , a contradiction. Thus R has at most three distinct maximal ideals. (cid:3) We end this paper with the following result.
Theorem 3.6
Let R be a ring. If (1) every proper ideal of R is weakly -absorbingprime, then either (2) R is a local ring with a unique maximal ideal M such that M = 0 or (3) R = R × R , where R is a local ring with a unique maximal ideal M suchthat M = 0 and R is a field or (4) R = R × R × R , where R , R , R are fields.Furthermore, (2) ⇒ (1) but (3) ⇒ (1) and (4) ⇒ (1) need not necessarily be true. Proof. If R satisfies condition (2), then the condition (1) follows from Theorem 3.2.Suppose R satisfies condition (3), i.e., R = R × R , where R is local with a uniquemaximal ideal M such that M = 0 and R is a field. Since R is a field and every properideal of R is a 1-absorbing prime, we conclude that the ideals { } × R and R × { } are weakly 1-absorbing prime ideals of R . Suppose that J is a non-zero proper ideal of R . It follows from Theorem 2.17 that J × R is a weakly 1-absorbing prime ideal, since J is a 1-absorbing prime ideal of R , by Corollary 3.3. Hence, it is enough to provethat J × { } is a weakly 1-absorbing prime ideal of R . Let 0 = a ∈ J . Then (0 , , =( a, a, ,
0) = ( a , , ∈ J ×{ } and both ( a , / ∈ J ×{ } and (1 , / ∈ J ×{ } . Thusthe condition (3) ⇒ (1) need not necessarily be true. Suppose that R satisfies condition(4), i.e., R = R × R × R , where R , R , R are fields. By taking I = R ×{ }×{ } , where R is a field, one can easily see that the ideal I is not weakly 1-absorbing prime, because(0 , , = (1 , , , , , ,
0) = (1 , , ∈ I and neither (1 , , , ,
1) = (1 , , ∈ I nor (1 , , ∈ I , showing that the condition (4) ⇒ (1) need not necessarily be true. To15omplete the proof, assume that every proper ideal of R is weakly 1-absorbing prime. Itfollows from Lemma 3.5 part (2) that R has at most three maximal ideals, and so if R is local with a unique maximal M , then by Theorem 3.2, M = { } . Thus R satisfies(2). If R has two maximal ideals M and M , then J ( R ) = M ∩ M is a weakly 1-absorbing prime ideal of R . Since every proper ideal of R is weakly 1-absorbing prime, Rabc is weakly 1-absorbing prime, for every a, b, c ∈ J ( R ). Hence abc = 0, by Lemma3.1, and this means that J ( R ) = M ∩ M = { } . Thus, by the Chinese RemainderTheorem R ∼ = R/M × R/M . But R/M and R/M are local rings and every properideal of R is weakly 1-absorbing prime. Hence, by Theorem 3.4, either R/M or R/M is a field and N = J = 0, where N and J are the maximal ideals of R/M and R/M ,respectively. Thus R satisfies (3). If R has three maximal ideals M , M and M , then J ( R ) = M ∩ M ∩ M is a weakly 1-absorbing prime ideal of R (weakly 2-absorbing ideal).But J ( R ) is not a 2-absorbing, hence, by [3, Theorem 2.4], J ( R ) = M ∩ M ∩ M = { } .Thus, by the Chinese Remainder Theorem, R ∼ = R/M × R/M × R/M . Since everyproper ideal of R is weakly 1-absorbing prime, Lemma 3.5 implies that R/M , R/M and R/M are fields. Thus R satisfies (4). (cid:3) References [1] D. D. Anderson, E. Smith, Weakly prime ideals, Houston J. Math. 29 (2003) 831–840.[2] A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc. 75 (2007)417–429.[3] A. Badawi, A. Y. Darani, On weakly 2-absorbing ideals of commutative rings, Houston J.Math. 39 (2013) 441–452.[4] A. Badawi, E. Yetkin, On 1-absorbing primary ideals of commutative rings, J. Algebra Appl.(2020), 2050111.[5] C. Beddani, W. Messirdi, 2-prime ideals and their applications, J. Algebra Appl. 15 (2016)1650051.[6] S. M. Bhatwadekar, P. K. Sharma, Unique factorization and birth of almost primes, Comm.Algebra 33 (2005) 43–49.[7] J. R. Hedstrom, E.G. Houston, Pseudo-valuation domains, Pacific J. Math. 75 (1978) 137–147.[8] J. A. Huckaba, Commutative Rings with Zero-Divisors, Marcel Dekker, New York/Basil, 1988.[9] R. Nikandish, M. J. Nikmehr, A. Yassine, More on the 2-prime ideals of commutative rings,Bull. Korean Math. Soc. 57 (2020) 117–126.
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