In this paper we propose a new concept of primeness&nbsp;in quantales. It is proved that this concept coincide with&nbsp;classical definition in commutative quantales, but no longer valid&nbsp;in the noncommutative setting. Also, the notions of strong and&nbsp;uniform strong primeness are investigated

Andrade, Alexsandra Oliveira

Bergamaschi, Flaulles Boone

Santiago, Regivan Hugo Nunes

English

Portal de Periódicos da UESB (Universidade Estadual do Sudoeste da Bahia)

Primeness in Quantales1st Flaulles Boone BergamaschiDept. de Ciˆencias Exatas e Tecnol´ogicasUniv. Estadual do Sudoeste da BahiaVitoria da Conquista, Brasilflaulles@yahoo.com.br2nd Regivan H. N. SantiagoDept. de Inf. e Matem´atica AplicadaUniv. Federal do Rio Grande do NorteNatal, Brasilregivan@dimap.ufrn.br3rd Alexsandra Oliveira AndradeDept. de Ciˆencias Exatas e Tecnol´ogicasUniv. Estadual do Sudoeste da BahiaVitoria da Conquista, Brasilalexsandra0andrade@gmail.comAbstract—In this paper we propose a new concept of prime-ness in quantales. It is proved that this concept coincide withclassical definition in commutative quantales, but no longer validin the noncommutative setting. Also, the notions of strong anduniform strong primeness are investigated.Keywords—Primeness; Prime ideal; Strong; Quantales.I. INTRODUCTIONAs it is known, ideals are the main object in the investiga-tion of ring theory and provide important information about therings because they are structural pieces. The same may occurin quantales. The definition of prime ideals proposed in [1] isbased on elements of a quantale and we ponder it is geared tocommutative environment. When we move from commutativeto the noncommutative setting, elementwise should be replacedby an approach based on ideals. Nevertheless, some authorsdefined the concept of primeness for commutative and non-commutative cases without realizing that this concept may notbe suitable for noncommutative setting as it was well shownby Navarro et. al. in [2]. We state that the concept of primeideal of general quantales could be defined as it is done in ringtheory, i.e. based on ideals.The main aim of this paper is to study the notion ofprimeness in the following perspective: we rename primeideal defined in [1] to completely prime ideal and definea new concept of prime ideal for general quantales. Thenwe translate an important result in ring theory for quantalesenvironment (theorem 2) to prove that these two conceptscoincide in the commutative setting, but are no longer valid inthe noncommutative setting (see Proposition 3). Also, basedon the studies of Lawrence and Handelman [3], started in1975, the notion of strong primeness is investigated for generalquantales.II. BASIC ON QUANTALESThis section proposes a new concept of prime idealssuitable for commutative and non-commutative quantales. Thedefinition of prime ideal used in [4] and [1] will be calledherein completely prime ideals. We drew attention to thetheorem 2 where prime ideals can be characterized in acertain way via elements. The Proposition 4 shows that inthe commutative case, prime and completely prime conceptscoincide, which are no longer valid in the noncommutativesetting according to Proposition 3. Finally, the concept ofquantale prime is proposed.Definition 1: [5] A quantale is a complete lattice Q withan associative binary operation   satisfying:a  _k2Kbk!=_k2K(a bk), _k2Kak! b =_k2K(ak b)for all a, b, ak, bk2 Q and k 2 K, where K is a index set.A quantale Q is called commutative whenever a  b = b afor a, b 2 Q. In this paper we denote the least and greatestelements of a quantale by ? and > respectively. If there existsan element e in Q such that x   e = e   x = x for all x in Qthe quantale is called a quantale with identity. In this paperwe consider quantales with identity.Definition 2: [4] Let Q be a quantale. A non-empty subsetI ✓ Q is called a right ideal of Q if it satisfies the followingconditions:i) a, b 2 I implies a _ b 2 I;ii) for all a, b 2 Q, a 2 I and b  a imply b 2 I ,iii) for all x 2 Q and a 2 I , we have a   x 2 I .Similarly we may define left ideal replacing (iii) by: (iii’)for all x 2 Q and a 2 I , we have x   a 2 I . If I is both rightand left ideal of Q, we call I a two-sided ideal or simply anideal of Q.Clearly by (ii) ? 2 I . Also, the set of all ideals of Q isclosed under arbitrary intersections.In Q we denote the subset I   J = {i   j 2 Q : i 2I and j 2 J} and A_B = {a_b : a 2 A and b 2 B}. Sincethe operation   is associative, we have (A B) C = A (B C).Also, if A is an two-sided ideal, then A Q, Q A, Q A Q ✓A.As usual, _ induces an order relation  on Q by puttingx  y , x _ y = y. Moreover,  is a congruence i.e. forevery x, y, u, v 2 Q if x  y and u  v, then x   u  y   v.To prove this, we first observe that if w  z then, for anys 2 Q, s   w  s   z and w   s  z   s because z = w _ zimplies s   z = s   (w _ z) = (s   w) _ (s   z) and z   s =(w_ z)   s = (w   s)_ (z   s); now suppose x  y and u  v,then x   u  y   u and y   u  y   v. Hence, x   u  y   v bytransitivity.ReCiC- Revista de Ciência de Computação 1III. PRIMENESSIn what follows we propose a more general definitionof prime ideals which encompasses commutative and non-commutative quantales.Definition 3: A prime ideal in a quantale Q is any properideal P such that, whenever I, J are ideals of Q with I J ✓ P ,either I ✓ P or J ✓ P .Definition 4: A subset P of a quantale Q is called com-pletely prime ideal if x and y are two elements of Q suchthat their product x   y 2 I , then x 2 I or y 2 I .As we will see the concept of prime and completely primeideals are different. But in a commutative quantale this conceptis the same.Proposition 1: If P is completely prime, then P is prime.Proof: Suppose that P is completely prime and I J ✓ P ,but J 6✓ P , where I, J are ideals of Q. Thus, there exists j 2 Jsuch that j /2 P . For all i 2 I we have i  j 2 I  J ✓ P , as Pis completely prime and j /2 P , then i 2 P . Therefore I ✓ P .The Proposition 3 will show that the converse of thisProposition is not true.Definition 5: [4] Let Q be a quantale and A ✓ Q. Theleast ideal containing A is called the ideal generated by A,and denoted as hAi.Clearly, h;i = {?}. If ; 6= A ✓ Q, then we have thefollowing result.Proposition 2: [4] Let A be a non-empty subset ofa quantale Q. Then hAi = {x 2 Q : x Wni=1 ai, for some positive integer n and a1, . . . , an 2 A [(A  Q) [ (Q  A) [ (Q  A  Q)}.We may denote hai = h{a}i and a  Q = {a}  Q.Lemma 1: hai  Q ✓ hai for all a 2 Q. If there exists anunit 1 in Q, then hai  Q = hai.Proof: Let x   q 2 hai   Q, where x 2 hai and q 2 Q.Hence, x q Wni=1(ai q), where ai q 2 a Q[Q a[Q a Q.Thus, x   q 2 hai. On the other hand if there exists unit 1 inQ, we write z 2 hai as z = z   1. Thus, z 2 hai  Q and wehave hai  Q = hai.Theorem 2: For an ideal P in Q the following statementsare equivalent:(1) P is prime ideal;(2) hai   hbi ✓ P implies a 2 P or b 2 P ;(3) a  Q   b ✓ P implies a 2 P or b 2 P .Proof: For (1) ) (2) note that hai and hbi are idealsof Q. As P is prime and hai   hbi ✓ P , then hai ✓ P orhbi ✓ P . Hence, a 2 P or b 2 P . For (2) ) (1), assume thatI  J ✓ P , but J 6✓ P , where I, J are ideals of Q. Thus, thereexists j 2 J such that j /2 P . Given i 2 I we have hii ✓ I .Hence, hii   hji ✓ I   J ✓ P . By hypothesis i 2 P or j 2 P .As j /2 P then we have i 2 P . Therefore, I ✓ P .For (3) ) (1), assume that I   J ✓ P , but J 6✓ P , whereI, J are ideals of Q. Thus, there exists j 2 J such that j /2 P .Given i 2 I we have i  Q   j ✓ I   J ✓ P . Hence, i 2 P orj 2 P , as j /2 P then we have i 2 P . Thus, I ✓ P .For (1) ) (3), suppose a  Q   b ✓ P , we first shall showthat hai  Q   hbi ✓ P . For this, let x   q   y 2 hai  Q   hbi,where x 2 hai, y 2 hbi and q 2 Q. Hence, by Proposition2, x  _ni=1ai and y  _mj=1bj , where ai 2 (a   Q) [ (Q  a) [ (Q   a   Q) and bi2 (b   Q) [ (Q   b) [ (Q   b   Q).Hence x   q   y  (_ni=1ai)   q   _mj=1bj = (_ni=1(ai   q))   _mj=1bj = _ni=1(ai q _mj=1bj) = _ni=1 _mj=1(ai   q   bj) .Observe that ai2 a Q[Q a[Q a Q and bj2 b Q[Q b[Q  b Q it is no hard to see that ai  q   bj2 a Q  b ✓ Pfor all i, j. As P is an ideal we have x   q   y 2 P . Thus,hai   Q   hbi ✓ P . By the Lemma 1 hai   Q = hai. Then,hai   Q   hbi = hai   hbi ✓ P . By the first proof ((1), (2))we have a 2 P or b 2 P .Proposition 3: There exists a noncommutative quantalewhere a prime ideal is not a completely prime ideal.Proof:Consider G the invertible 2 ⇥ 2 matrices under mul-tiplication over the real interval [0, 1] and the partial or-der A  B , aij bij. According to Rosenthal[5], page 19, example 16, any complete partially ordered group(written multiplicatively) is a quantale with a   b = a · b.Thus, G is a noncommutative quantale. Let h0i as an idealgenerated by 0, clearly h0i = {0}. We will show that theh0i (zero ideal) is prime, but not completely prime by usingthe Theorem 2 (3). Thus, suppose that X =✓a bc d◆andY =✓e fg h◆are two matrices such that X  G   Y ✓ h0i.Hence X   T   Y =✓0 00 0◆for all matrix T 2 G. Then,in particular,X  ✓1 00 0◆ Y =✓a bc d◆✓1 00 0◆✓e fg h◆=✓ae afce cf◆= 0 , a = c = 0 or e = f = 0,X✓0 10 0◆Y =✓a bc d◆✓0 10 0◆✓e fg h◆=✓ag ahcg ch◆= 0 , a = c = 0 or g = h = 0,X  ✓0 01 0◆ Y =✓a bc d◆✓0 01 0◆✓e fg h◆=✓be bfde df◆= 0 , b = d = 0 or e = f = 0,X  ✓0 00 1◆ Y =✓a bc d◆✓0 00 1◆✓e fg h◆=✓bg bhdg dh◆= 0 , b = d = 0 or g = h = 0,ReCiC- Revista de Ciência de ComputaçãoText 2Hence, a solution must verify that X =✓0 00 0◆orY =✓0 00 0◆. Therefore X 2 h0i or Y 2 h0i and then h0iis prime. Nevertheless, h0i is not completely prime, since✓0 10 0◆ ✓0 10 0◆=✓0 00 0◆although✓0 10 0◆/2 h0i.Proposition 4: In a commutative quantale an ideal is com-pletely prime iff it is prime.Proof: If P is a completely prime ideal of a quantaleQ, then by the Proposition 1 P is prime. On the other hand,suppose P is a prime ideal and a   b 2 P for any a, b 2 Q.Let x   y 2 hai   hbi, where x 2 hai and y 2 hbi. Thus,x   y Wni=1 ai  Wmj=1 bj =Wni=1⇣Wmj=1(ai   bj)⌘, whereai2 a Q[Q a[Q a Q and bj2 b Q[Q b[Q b Q. As Qis commutative a Q = Q a = Q a Q and b Q = Q b = Q b Q. Thus, ai bj2 a Q b Q = a b Q for all i = 1, . . . , nand j = 1, . . . ,m. Hence, ai  bj= a   b   q 2 P and thenx   y Wni=1⇣Wmj=1(ai   bj)⌘2 P . Therefore, hai   hbi ✓ Pand by the Theorem 2 we have a 2 P or b 2 P .In what follows, we will introduce the notion of quantaleprime. As we know, in ring theory, a quantale is prime iff theideal generated by 0 is a prime ideal. Then, the Proposition 5translates this result into quantale environment.Definition 6: A quantale Q is called prime if given a, b 2Q with a 6= ? and b 6= ?, there exists f 2 Q such thata   f   b 6= ?.Proposition 5: A quantale Q is prime iff h?i is a primeideal.Proof: Suppose Q prime and assume that I   J ✓ h?i,but I, J 6✓ h?i, where I, J are ideals of Q. Thus, there existsi 2 I ,j 2 J such that i, j 6= ?. As Q is prime, there existsq 2 Q such that i   q   j 6= ?, then we have a contradictionbecause i   q   j 2 I   J ✓ h?i. Hence, I ✓ h?i or J ✓ h?i.On the other hand, suppose h?i is a prime ideal of Q. Givena, b 6= ? in Q, suppose a   q   b = ? for all q 2 Q. Hence,a   Q   b ✓ h?i. As h?i is a prime ideal, then a 2 h?i orb 2 h?i, but a, b 6= ?.IV. STRONG PRIMENESS IN QUANTALESStrongly prime rings were introduced in 1973, as a primering with finite condition in the generalization of results ongroup rings proved by Lawrence in [3]. In this section we bringthis concept to quantales making specific adaptations for thisenvironment.Definition 7: Let A be a subset of a quantale Q. The rightannihilator of A is defined as Anr(A) = {x 2 Q : Ax =h?i}. Similarly, we can define the left annihilator Anl.Definition 8: [3] A quantale Q is called right stronglyprime if for each x 2 Q  {?} there exists a finite nonemptysubset Fxof Q such that Anr(x   Fx) = h?i. Clearly if Q isright strongly prime, then Q is prime. The set Fxis called aninsulator of x in Q.Proposition 6: If Q is right strongly prime, then everynonzero ideal I of Q contains a finite subset F which hasright annihilator zero.Proof: Suppose Q right strongly prime. Let x 2 I andx 6= ? and F = x   Fx✓ I . Thus, Anr(F ) = h?i.It is clear that every right strongly prime quantale is a primequantale. It is also possible to define left strongly prime in asimilar manner for right strong primeness.Definition 9: A quantale Q is called uniformly stronglyprime (usp) if the same right insulator may be chosen for eachnonbottom element.Proposition 7: A quantale Q is a right uniformly stronglyprime iff there exists a finite subset F ✓ Q such that for anytwo nonbottom elements x and y of Q, there exists f 2 Fsuch that x   f   y 6= ?.Proof: Let Q be uniformly right strongly prime quantale.Hence Q has a uniform right insulator F which is a finite setsuch that for any element x 2 Q, x F has no nonbottom rightannihilators. Thus, if x and y are any two nonbottom elementsin Q, y cannot be in the annihilator of x   F . Hence there isan f 2 F such that x   f   y 6= ?. For the reverse implicationit is easy to see that if the condition is satisfied then for anyx 6= ? in Q, no nonbottom element annihilates x   F on theright. Hence Q is uniformly right strongly primeIt is clear that the condition in Proposition 7 is notone-sided; consequently, this condition is also equivalent touniformly left strongly prime, and we have:Corollary 3: Q is uniformly right strongly prime iff Q isuniformly left strongly prime.Corollary 4: A quantale Q is uniformly strongly prime iffthere exists a finite subset F ✓ Q such that a   F   b = ?implies a = ? or b = ? for all a, b 2 Q.Proof: Straightforward.Definition 10: An ideal P 6= h?i of a quantale Q is calleduniformly strongly prime (usp) ideal if there exists a finitesubset F ✓ Q such that a F  b ✓ P implies a 2 P or b 2 P .Proposition 8: An ideal I of a quantale Q is a usp idealiff there exists a finite subset F ✓ Q such that for any twoelements a, b 2 Q \ I(complement of I in Q), there existsf 2 F such that x   f   y /2 I .Proof: Suppose I a usp ideal of Q. If a /2 I and b /2 Iby Definition 10 a   F   b is not a subset of I . Hence, thereexists f 2 F such that a   f   b /2 I . For the converse, notethat by hypothesis it is impossible to have a   F   b ✓ I anda /2 I and b /2 I .ReCiC- Revista de Ciência de Computação 3V. FINAL REMARKSPrime ideals have developed an important role in ringtheory and have attracted the attention of some researchers inthe investigation of quantales. As prime ideals are structuralpieces of a ring it is relevant to study its concept in order toestablish a well-founded quantale theory. With this in mind, itis necessary to investigate primeness over arbitrary quantales,i.e. commutative and noncommutative setting. Therefore, thispaper invites the reader to think about primeness in quantalesbringing a new perspective for prime ideals.REFERENCES[1] Lingyun Yang and Luoshan Xu. Roughness in quantales. InformationSciences, 220:568–579, 2013.[2] Gabriel Navarro, Oscar Cortadellas, and F.J. Lobillo. Prime fuzzy idealsover noncommutative rings. Fuzzy Sets and Systems, 199(0):108 – 120,2012.[3] David Handelman and John Lawrence. Strongly prime rings. Transac-tions of the American Mathematical Society, 211:209–223, 1975.[4] Qingjun Luo and Guojun Wang. Roughness and fuzziness in quantales.Information Sciences, 271:14–30, 2014.[5] Kimmo I Rosenthal. Quantales and their applications, volume 234.Longman Scientific & Technical Harlow, 1990.ReCiC- Revista de Ciência de Computação 4

Primeness in Quantales

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Primeness in Quantales

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