Flat topology on prime, maximal and minimal prime spectra of quantales
aa r X i v : . [ m a t h . L O ] J un Flat topology on prime, maximal and minimalprime spectra of quantales
George Georgescu
University of BucharestFaculty of Mathematics and Computer ScienceBucharest, RomaniaEmail: [email protected]
Abstract
Several topologies can be defined on the prime, the maximal and theminimal prime spectra of a commutative ring; among them, we mentionthe Zariski topology, the patch topology and the flat topology. By usingthese topologies, Tarizadeh and Aghajani obtained recently new charac-terizations of various classes of rings: Gelfand rings, clean rings, absolutelyflat rings, mp - rings,etc. The aim of this paper is to generalize some oftheir results to quantales, structures that constitute a good abstractizationfor lattices of ideals, filters and congruences. We shall study the flat andthe patch topologies on the prime, the maximal and the minimal primespectra of a coherent quantale. By using these two topologies one obtainsnew characterization theorems for hyperarchimedean quantales, normalquantales, B-normal quantales, mp - quantales and P F - quantales. Thegeneral results can be applied to several concrete algebras: commutativerings, bounded distributive lattices, MV-algebras, BL-algebras, residuatedlattices, commutative unital l - groups, etc. Keywords : flat topology, coherent quantale, reticulation, hyperarchimedeanquantales, normal and B - normal quantales, mp - quantales. The flat topology on the prime spectrum of a ring was introduced by Hochsterin [22] under the name of inverse topology. It was rediscovered by Doobs et al.in [14], where the terminology of ”flat topology” appeared. The flat topology isstrongly related to other two topologies defined on the prime spectrum of a com-mutative ring: spectral topology and patch topology [13],[25],[43]. These threetopologies have a deep impact on some important themes in ring theory (see[22],[13],[1]). Recently, Tarizadeh proposed in [43] purely algebraic definitionsfor flat and patch topologies on the prime spectrum
Spec ( R ) of a commutativering R . For examples, the closed sets in the flat topology on Spec ( R ) are defined1s the images Im ( f ∗ ), where f ∗ : Spec ( A ) → Spec ( R ) is the map induced bya flat morphism f : R → A . Further, the flat and the patch topologies wereused to obtain new properties and new characterizations of Gelfand rings andclean rings, as well as of new classes of rings: mp-rings and purified rings (see[1],[44],[45]). A natural problem is to use the flat and the patch topology forobtaining new results on the spectra of other types of algebras. On the otherhand, the quantales constitute a good abstraction of the lattices of ideals, filtersand congruences in various algebraic structures. A rich literature was dedicatedto the spectra of quantales (see [15],[17],[35],[36],[38]). Besides the quantales,other types of multiplicative lattices were proposed to abstractize the lattices ofideals, filters and congruences [19], [30],[40], [41]. In this paper we shall studythe flat topology on prime, maximal and minimal prime spectra of a coherentquantale in connection to important classes of quantales: hyperarchimedean,normal, B -normal, mp - quantales, etc. We shall obtain new characterizationsof these types of quantales in terms of some algebraic and topological propertiesof spectra. Our results extend some theorems proven in [1],[43],[44],[45], [3],[4], etc. for the spectra of commutative rings. The proofs of some results willuse the reticulation of a quantale, a construction that assigns to each coherentquantale A a unique bounded distributive lattice L ( A ), whose prime spectrum Spec Id ( L ( A )) is homeomorphic to the prime spectrum of A .Now we shall describe the content of this paper. In Section 2 we present somenotions and basic results on the prime and the maximal spectra of a quantale,their spectral topologies and the radical elements (cf. [38],[30],[15],[36]).Section 3 contains the axiomatic definition of reticulation of a coherent quan-tale A and the connection between the m -prime elements of A and the primeideals of the reticulation L ( A ). The homeomorphism between the prime spec-trum Spec ( A ) of A and the space Spec Id ( L ( A )) of the prime ideals in L ( A )(with the Stone topology) induces on Spec ( A ) a spectral topology (this spectralspace is denoted by Spec Z ( A ), because it generalizes the Zariski topology).Section 4 deals with the Boolean center B ( A ) of a quantale A . We presenta short proof that B ( A ) is isomorphic to the Boolean algebra B ( L ( A )) of com-plemented elements in L ( A ). Section 5 concerns the patch and the flat topologyon Spec ( A ), associated with the spectral space Spec Z ( A );these two topologicalspaces will be denoted by Spec P ( A ), resp. Spec F ( A ). We also introduce thePierce spectrum Sp ( A ) of the quantale A . In Section 6 we continue the study ofthe hyperarchimedean quantales, initiated in [9]. The main result of the sectionpresents new characterizations of these objects in terms of the flat and the patchtopologies.Section 7 contains results about the flat topology on the maximal spectrum M ax ( A ) of the coherent quantale A . The new topological space M ax F ( A ) isHausdorff and zero - dimensional. We use the flat topology on M ax ( A ) in orderto obtain new properties that characterize the normal and the B-normal quan-tales [9],[20],[35],[41]. The normal quantales constitute an abstractization of thelattices of ideals in Gelfand rings [25],[27],[32] and in normal lattices[11],[37],[19],while the B-normal quantales generalize the lattices of ideals in clean rings[34],[23] and in B-normal lattices [10],[8]. Thus our theorems on normal and2-normal quantales can be applied to various types of structures: Gelfand ringsand normal lattices, normal and B - normal lattices, Gelfand residuated latticesand residuated lattices with Boolean lifting property [18], clean unital l - groups[21],etc.In Section 8 we study two topologies on the set M in ( A ) of the minimal m - prime elements of a coherent quantale A . Thus we obtain two topologicalspaces M in Z ( A ) and M in F ( A ) : the first one is a subspace of SpecZ ( A ) and thesecond one is a subspace of Spec F ( A ). By using the reticulation we prove that M in Z ( A ) is a zero - dimensional Hausdorff space and M in F ( A ) is a compact T mp - quantales, a notion thatgeneralizes the mc - rings of [1]. We present several conditions that characterize mp - quantales. For example, we prove that a coherent quantale A is an mc - quantale iff the reticulation L ( A ) is a conormal lattice [40] iff Spec F ( A ) is anormal space. We introduce the P F - quantales as a generalization of the
P F - rings [4] and we prove that a coherent quantale A is a P F - quantale if andonly if it is a semiprime mp - quantale. Further we obtain a characterizationtheorem for P F - quantales.
Let ( A, ∨ , ∧ , · , ,
1) be a quantale and K ( A ) the set of its compact elements. A is said to be integral if ( A, · ,
1) is a monoid and commutative, if the multi-plication · is commutative. A frame is a quantale in which the multiplicationcoincides with the meet [25]. The quantale A is algebraic if any a ∈ A has theform a = W X for some subset X of K ( A ). An algebraic quantale A is coherentif 1 ∈ K ( A ) and K ( A ) is closed under the multiplication. Throughout this pa-per, the quantales are assumed to be integral and commutative. Often we shallwrite ab instead of a · b . We fix a quantale A . Lemma 2.1 [7] For all elements a, b, c of the quantale A the following hold:(1) If a ∨ b = 1 then a · b = a ∧ b ;(2) If a ∨ b = 1 then a n ∨ b n = 1 for all integer numbers n ≥ ;(3) If a ∨ b = a ∨ c = 1 then a ∨ ( b · c ) = a ∨ ( b ∧ c ) = 1 ;(4) If a ∨ b = 1 and a ≤ c then a ∨ ( b · c ) = c . One can define on the quantale A a residuation operation a → b = W { x | ax ≤ b } and a negation operation a ⊥ = a → W { x | ax = 0 } . Thus ( A, ∨ , ∧ , · , → , ,
1) is a residuation lattice [16], [26]. In this paper we shall use withoutmention the basic arithmetical properties of a residuated lattice.An element p < A is m - prime if for all a, b ∈ A , ab ≤ p implies a ≤ b or b ≤ p . If A is an algebraic quantale, then p < m -prime if and onlyif for all c, d ∈ K ( A ), cd ≤ p implies c ≤ p or d ≤ p . Let us introduce thefollowing notations: Spec ( A ) is the set of m -prime elements and M ax ( A ) is the3et of maximal elements of A . If 1 ∈ K ( A ) then for any a < m ∈ M ar ( A ) such that a ≤ m . The same hypothesis 1 ∈ K ( A ) implies that M ax ( A ) ⊆ Spec ( A ).Let R be a (unital) commutative ring and L a bounded distributive lattice.Let us denote by Id ( R ) the quantale of ideals in R and by Id ( L ) the frame ofideals in L . Thus the set Spec ( R ) of prime ideals in R is the prime spectrumof the quantale Id ( R ) and the set of prime ideals in L is the prime spectrum ofthe frame Id ( L ).The radical ρ ( a ) = ρ A ( a ) of an element a ∈ A is defined by ρ A ( a ) = V { p ∈ Spec ( A ) | a ≤ p } ; if a = ρ ( a ) then a is a radical element. We shall denote by R ( A ) the set of radical elements of A . The quantale is semiprime if ρ (0) = 0. Lemma 2.2 [38] For all elements a, b ∈ A the following hold:(1) a ≤ ρ ( a ) ;(2) ρ ( a ∧ b ) = ρ ( ab ) = ρ ( a ) ∧ ρ ( b ) ;(3) ρ ( a ) = 1 iff a = 1 ;(4) ρ ( a ∨ b ) = ρ ( ρ ( a ) ∨ ρ ( b )) ;(5) ρ ( ρ ( a )) = ρ ( a ) ;(6) ρ ( a ) ∨ ρ ( b ) = 1 iff a ∨ b = 1 ;(7) ρ ( a n ) = ρ ( a ) , for all integer n ≥ . For an arbitrary family ( a i ) i ∈ I ⊆ A , the following equality holds: ρ ( _ i ∈ I a i ) = ρ ( _ i ∈ I ρ ( a i )). If ( a i ) i ∈ I ⊆ R ( A ) then we denote · _ i ∈ I a i = ρ ( _ i ∈ I a i ). Thus it easyto prove that ( R ( A ) , · _ , ∧ , ρ ( a ) ,
1) is a frame [38].
Lemma 2.3 [9] If ∈ K ( A ) then Spec ( A ) = Spec ( R ( A )) and M ax ( A ) = M ax ( R ( A )) . Lemma 2.4 [30] Let A be a coherent quantale and a ∈ A . Then(1) ρ ( a ) = W { c ∈ K ( A ) | c k ≤ a for some integer k ≥ } ;(2) For any c ∈ K ( A ) , c ≤ ρ ( a ) iff c k ≤ a for some k ≥ . Lemma 2.5 [9] If A is a coherent quantale then K ( R ( A )) = ρ ( K ( A )) and R ( A ) is a coherent frame. For any element a of a coherent quantale A let us consider the interval[ a ) A = { x ∈ A | a ≤ x } and for all x, y ∈ [ a ) A denote x · a y = xy ∨ a . Thus[ a ) A is closed under the multiplication · a and ([ a ) A , ∨ , ∧ , · a , ,
1) is a coherentquantale. 4 emma 2.6 [9] The quantale ([ ρ ( a )) A , ∨ , ∧ , · a , , is semiprime and Spec ( A ) = Spec ([ ρ ( a )) A ) , M ax ( A ) = M ax ([ ρ ( a )) A ) . Let
A, B be two quantales. A function f : A → B is a morphism of quan-tales if it preserves the arbitrary joins and the multiplication; f is an integralmorphism if f (1) = 1. Lemma 2.7 [9] Let A be a coherent quantale and a ∈ A .(1) The function u Aa : A → [ a ) A , defined by u Aa ( x ) = x ∨ a , for all x ∈ A ,is an integral quantale morphism;(2) If c ∈ K ( A ) then u Aa ( c ) ∈ K ([ a )) . Let A be a quantale such that 1 ∈ K ( A ). For any a ∈ A , denote D ( a ) = { p ∈ Spec ( A ) | a p } and V ( a ) = { p ∈ Spec ( A ) | a ≤ p } . Then Spec ( A ) isendowed with a topology whose closed sets are ( V ( a )) a ∈ A . If the quantale A isalgebraic then the family ( D ( c )) c ∈ K ( A ) is a basis of open sets for this topology.The topology introduced here generalizes the Zariski topology (defined on theprim spectrum Spec ( R ) of a commutative ring R [2]) and the Stone topology(defined on the prime spectrum Spec Id ( L ) of a bounded distributive lattice L [5]).Thus we denote by Spec Z ( A ) the prime spectrum Spec ( A ) endowed with theabove defined topology; M ax Z ( A ) will denote the maximal spectrum M ax ( A )considered as a subspace of Spec Z ( A ).Let L be a bounded distributive lattice. For any x ∈ L , denote D Id ( x ) = { P ∈ Spec Id ( L ) | x P } and V Id ( x ) = { P ∈ Spec
Id,Z ( L ) | x ∈ P } . The family( D Id ( x )) x ∈ L is a basis of open sets for the Stone topology on Spec Id ( L ); thistopological space will be denoted by Spec
Id,Z ( L ). Let M ax Id ( L ) be the set ofmaximal ideals of L . Thus M ax Id ( L ) ⊆ Spec Id ( L ) and M ax Id ( L ) becomes asubspace of Spec Id ( L ), denoted M ax
Id,Z ( L ). In this section we shall recall from [9],[17] the axiomatic definition of thereticulation of the coherent quantale and some of its basic properties. Let A bea coherent quantale and K ( A ) the set of its compact elements. Definition 3.1 [9] A reticulation of the quantale A is a bounded distributivelattice L together a surjective function λ : K ( A ) → L such that for all a, b ∈ K ( A ) the following properties hold(1) λ ( a ∨ b ) ≤ λ ( a ) ∨ λ ( b ) ;(2) λ ( ab ) = λ ( a ) ∧ λ ( b ) ;(3) λ ( a ) ≤ λ ( b ) iff a n ≤ b , for some integer n ≥ .
5n [9],[17] there were proven the existence and the unicity of the reticu-lation for each coherent quantale A ; this unique reticulation will be denotedby ( L ( A ) , λ A : K ( A ) → L ( A )) or shortly L ( A ). The reticulation L ( R ) of acommutative ring R there was introduced by many authors, but the main ref-erences on this topic remain [40], [25]. We remark that L(R) is isomorphic tothe reticulation L(Id(R)) of the quantale Id ( R ). Lemma 3.2 [9] For all elements a, b ∈ K ( A ) the following properties hold:(1) a ≤ b implies λ A ( a ) ≤ λ A ( b ) ;(2) λ A ( a ∨ b ) = λ A ( a ) ∨ λ A ( b ) ;(3) λ A ( a ) = 1 iff a = 1 ;(4) λ A (0) = 0 ;(5) λ A ( a ) = 0 iff a n = 0 , for some integer n ≥ ;(6) λ A ( a n ) = λ A ( a ) , for all integer n ≥ ;(7) ρ ( a ) = ρ ( b ) iff λ A ( a ) = λ A ( b ) ;(8) λ A ( a ) = 0 iff a ≤ ρ (0) ;(9) If A is semiprime then λ A ( a ) = 0 implies a = 0 . For any a ∈ A and I ∈ Id ( L ( A )) let us denote a ∗ = { λ A ( c ) | c ∈ K ( A ) , c ≤ a } and I ∗ = W { c ∈ K ( A ) | λ A ( c ) ∈ I } . Lemma 3.3 [9] The following assertions hold(1) If a ∈ A then a ∗ is an ideal of L ( A ) and a ≤ ( a ∗ ) ∗ ;(2) If I ∈ Id ( L ( A )) then ( I ∗ ) ∗ = I ;(3) If p ∈ Spec ( A ) then ( p ∗ ) ∗ = p and p ∗ ∈ Spec Id ( L ( A )) ;(4) If P ∈ Spec Id (( L ( A )) then P ∗ ∈ Spec ( A ) ;(5) If p ∈ K ( A ) then c ∗ = ( λ A ( c )] . Lemma 3.4 [9] If a ∈ A and I ∈ Id ( L ( A )) then ρ ( a ) = ( a ∗ ) ∗ , a ∗ = ( ρ ( a )) ∗ and ρ ( I ∗ ) = I ∗ . Lemma 3.5 If c ∈ K ( A ) and I ∈ Id ( L ( A )) then c ≤ I ∗ iff λ A ( c ) ∈ I . Proof. If c ≤ W { d ∈ K ( A )) | λ A ( d ) ∈ I } then there exists d ∈ K ( A ) suchthat λ A ( c ) ∈ I and c ≤ d . Thus λ A ( c ) ≤ λ A ( d ), so λ A ( c ) ∈ I . The converseimplication is obvious. 6 emma 3.6 Assume that c ∈ K ( A ) and p ∈ Spec ( A ) . Then c ≤ p iff λ A ( c ) ∈ p ∗ . According to Lemma 3.3, one can consider the following order- preserving func-tions: u : Spec ( A ) → Spec Id ( L ( A )) and v : Spec Id ( L ( A )) → Spec ( A ), definedby u ( p ) = p ∗ and v ( P ) = P ∗ , for all p ∈ Spec ( A ) and P ∈ Spec Id ( L ( A )).Sometimes the previous functions u and v will be denoted by u A and v A . Lemma 3.7 [9] The functions u and v are homeomorphisms, inverse to oneanother. Corollary 3.8
M ax Z ( A ) and M ax
Id,Z ( L ( A )) are homeomorphic. Proposition 3.9 [9] The functions
Φ : R ( A ) → Id ( L ( A )) and Ψ : Id ( L ( A )) → R ( A ) defined by Φ( a ) = a ∗ and Ψ( I ) = I ∗ , for all a ∈ R ( A ) and I ∈ Id ( L ( A )) ,are frame isomorphisms, inverse to one another. Corollary 3.10 If I, J are ideals of L ( A ) then ( I ∨ J ) ∗ = ρ ( I ∗ ∨ J ∗ ) . Proof.
The equality ( I ∨ J ) ∗ = ρ ( I ∗ ∨ J ∗ ) follows from the fact that the frameisomorphism Ψ preserves the finite joins. The Boolean center of a quantale A is the Boolean algebra B ( A ) of comple-mented elements of A (cf. [7],[24]). In this section we shall study some basicproperties of the Boolean center of a coherent quantale versus the reticulation. Lemma 4.1 [7],[24] Let A be a quantale and a, b ∈ A , e ∈ B ( A ) . Then thefollowing properties hold:(1) a ∈ B ( A ) iff a ∨ a ⊥ = 1 ;(2) a ∧ b = ae ;(3) e → a = e ⊥ ∨ a ;(4) If a ∨ b = 1 and ab = 0 , then a, b ∈ B ( A ) ;(5) ( a ∧ b ) ∨ e = ( a ∨ e ) ∧ ( b ∧ e ) ;(6) For any integer n ≥ , a ∨ b = 1 and a n b n = 0 implies a n , b n ∈ B ( A ) . Proof.
The properties (1)-(5) are taken from [7],[24] and (6) follows by (4) andLemma 2.1,(ii). 7 emma 4.2 [9] If ∈ K ( A ) then B ( A ) ⊆ K ( A ) . For a bounded distributive lattice L we shall denote by B ( L ) the Boolean algebraof the complemented elements of L . It is well-known that B ( L ) is isomorphicto the Boolean center B ( Id ( L )) of the frame Id ( L ) (see [7], [25], [8]).Let us fix a coherent quantale A . Lemma 4.3
Assume c ∈ K ( A ) . Then λ A ( c ) ∈ B ( L ( A )) if and only if c n ∈ B ( A ) , for some integer n ≥ . Proof.
Assume λ A ( c ) ∈ B ( L ( A )), hence λ A ( c ) ∨ λ A ( d ) = 1 and λ A ( c ) ∧ λ A ( d ) =0, for some d ∈ K ( A ). Then λ A ( c ∨ d ) = 1 and λ A ( cd ) = 0, hence, by Lemma3.2,(2) and (5), it follows that c ∨ d = 1 and c n d n = 0, for some integer n ≥ c n , d n ∈ B ( A ). Conversely, if c n ∈ B ( A )then λ A ( c ) = λ A ( c n ) is an element of B ( L ( A )). Corollary 4.4 [9] The function λ A | B ( A ) : B ( A ) → B ( L ( A )) is a Boolean iso-morphism. Proof.
It is easy to see that the function λ A | B ( A ) : B ( A ) → B ( L ( A )) is aninjective Boolean morphism. The surjectivity follows by using Lemma 4.3.If L is bounded distributive lattice and I ∈ Id ( L ) then the annihilator of I is the ideal Ann ( I ) = { x ∈ L | x ∧ y = 0, for all y ∈ L } .The next two propositions concern the behaviour of reticulation w.r.t. theannihilators. Proposition 4.5 If a is an element of a coherent quantale then Ann ( a ∗ ) =( a → ρ (0)) ∗ ; if A is semiprime then Ann ( a ∗ ) = ( a ⊥ ) ∗ . Proof.
Assume x ∈ Ann ( a ∗ ), so x = λ A ( c ) for some c ∈ K ( A ) with theproperty that for all d ∈ K ( A ), d ≤ a implies λ A ( cd ) = λ A ( c ) ∧ λ A ( d ) = 0. ByLemma 3.2(8) one gets cd ≤ ρ (0), so c ≤ d → ρ (0). Thus the following hold: c ≤ V { d → ρ (0) | d ∈ K ( A ) , d ≤ a } = ( W { d ∈ K ( A ) | d ≤ a } ) → ρ (0) = a → ρ (0),hence x = λ A ( c ) ∈ ( a → ρ (0)) ∗ . We conclude that Ann ( a ∗ ) ⊆ ( a → ρ (0)) ∗ .In order to prove that ( a → ρ (0)) ∗ ⊆ Ann ( a ∗ ) assume that x ∈ ( a → ρ (0)) ∗ ,so x = λ A ( c ) for some c ∈ K ( A ) such that c ≤ a → ρ (0). For all d ∈ K ( A )with d ≤ a we have c ≤ a → ρ (0) ≤ d → ρ (0). By Lemma 3.2,(8) one gets λ A ( c ) ∧ λ ( d ) = λ A ( cd ) = 0, so x = λ A ( c ) ∈ Ann ( a ∗ ). Proposition 4.6
Assume that A is a coherent quantale. If I is an ideal of L ( A ) then ( Ann ( I )) ∗ = I ∗ → ρ (0) ; if A is semiprime then ( Ann ( I )) ∗ = ( I ∗ ) ⊥ . Proof.
In order to verify that I ∗ → ρ (0) ≤ ( Ann ( I )) ∗ , it suffices to show thatfor all c ∈ K ( A ), c ≤ I ∗ → ρ (0) implies c ≤ ( Ann ( I )) ∗ . If c ≤ I ∗ → ρ (0) then8 { cd | d ∈ K ( A ) , λ A ( d ) ∈ I } = c ( W { d ∈ K ( A ) | λ A ( d ) ∈ I } ) = cI ∗ ≤ ρ (0).Thus for all d ∈ K ( A ) with λ A ( d ) ∈ I we have cd ≤ ρ (0) so c n d n = 0 for someinteger n ≥ λ A ( c ) ∧ λ A ( d ) = λ A ( c n d n ) = 0,hence λ A ( c ) ∈ Ann ( I ), i.e. c ≤ ( Ann ( I )) ∗ .Assume now that c ∈ K ( A ) and c ≤ ( Ann ( I )) ∗ , hence by Lemma 3.5, λ A ( c ) ∈ Ann ( I ). For any c ∈ K ( A ) with λ A ( d ) ∈ I we have λ A ( cd ) = λ A ( c ) ∧ λ A ( d ) = 0, hence by Lemma 3.2 (8) one gets cd ≤ ρ (0). Therefore we have cI ∗ = W { cd | d ∈ K ( A ) , λ A ( d ) ∈ I } ≤ ρ (0), i.e. c ≤ I ∗ → ρ (0). Then theinequality ( Ann ( I )) ∗ ≤ I ∗ → ρ (0) is proven, so the equality ( Ann ( I )) ∗ = I ∗ → ρ (0) follows.An element a of an arbitrary quantale A is said to be pure (or virginal, inthe terminology of [20]) if for all c ∈ K ( A ), c ≤ a implies a ∨ c ⊥ = 1. The pureelements in a quantale extend the pure ideals of a ring [27],[41] and the σ -idealsof a bounded distributive lattice [12], [19]. More precisely, an ideal I of boundeddistributive lattice L is a σ - ideal if for all x ∈ I , we have I ∨ Ann ( x ) = L . Lemma 4.7
If an element a of coherent quantale A is pure then a ∗ is a σ - idealof the reticulation L ( A ) . If moreover A is semiprime then for each σ - ideal J of L ( A ) , J ∗ is a pure element of A . Proof.
Assume x ∈ a ∗ , so x = λ A ( c ) for some c ∈ K ( A ) with c ≤ a . Since a is pure, c ≤ a implies a ∨ c ⊥ = 1, so d ∨ e = 1 for some d, e ∈ K ( A ) withthe properties d ≤ a and e ≤ c ⊥ . Thus λ A ( d ) ∈ a ∗ and λ A ( c ) ∧ λ A ( e ) = λ A ( ce ) = λ (0) = 0, i.e. λ A ( e ) ∈ Ann ( λ A ( c )). We observe that λ A ( d ) ∨ λ A ( e ) = λ A ( d ∨ e ) = 1, so a ∗ ∨ Ann ( λ A ( e )) = L ( A ). Thus a ∗ is a σ - ideal.Now we assume that A is semiprime and J is σ - ideal of L ( A ). In order toprove that J ∗ is a pure element of A let us consider a compact element c of A such that c ≤ J ∗ . By Lemma 3.5 we have λ A ( c ) ∈ J , hence J ∨ Ann ( λ A ( c )) = L ( A ), so there exist two compact elements d and e of A such that λ A ( d ) ∈ J , λ A ( e ) ∈ Ann ( λ A ( c )) and λ A ( d ∨ e ) = λ A ( d ) ∨ λ A ( e ) = 1. According to Lemmas3.5 and 3.2,(3) we get d ≤ J ∗ and d ∨ e = 1. From λ A ( e ) ∈ Ann ( λ A ( c )) weinfer λ A ( ce ) = λ A ( c ) ∧ λ A ( e ) = 0, hence ce = 0 (because A is semiprime). Thus e ≤ c ⊥ , therefore 1 = d ∨ e ≤ J ∗ ∨ c ⊥ . It follows that J ∗ ∨ c ⊥ = 1, hence J is a σ - ideal of L ( A ). In this section we shall discuss some basic properties concerning three topolo-gies defined on the prime spectrum
Spec ( A ) of a coherent quantale A : spectraltopology, flat topology and patch topology. A topological space ( X, Ω) is saidto be spectral [22] ( or coherent in the terminology of [25]) if it is sober and the9amily K (Ω) of compact open sets of X is closed under finite intersections, andforms a basis for the topology. The standard examples of spectral spaces arethe prime spectrum Spec ( A ) of a commutative ring R (with the Zariski topol-ogy) and the prime spectrum Spec Id ( L ) of bounded distributive lattice L (withthe Stone topology). If ( X, Ω) is a spectral space then in a standard way (see[13],[25]) one can define on X the following two topologies: • the patch topology, having as basis the family of sets U S V , where U isa compact open set in X and V is the complement of a compact open set (thistopological space is denoted by X P ); • the flat topology, having as basis the family of the complements of compactopen sets in X (this topological space is denoted by X F ). Lemma 5.1 [15], [25] X P is a Boolean space and X F is a spectral space. Remark 5.2 If L is a bounded distributive lattice and X is the spectral space Spec Id ( L ) then the family ( D Id ( x ) T V Id ( y )) x,y ∈ L is a basis of open sets for X P and the family ( V Id ( y )) y ∈ L is a basis of open sets for X F . Let A be a coherent quantale. By Proposition 3.7, Spec Z ( A ) is homeomor-phic with the spectral space Spec Id ( L ( A )), hence it is a spectral space. Thefamily of open sets in Spec Z ( A ) will be denoted by Z = Z A . For any sub-set S of Spec ( A ), cl Z ( S ) = V ( T S ) is the closure of S in Spec Z ( A ); for all p ∈ Spec ( A ), we have cl Z ( { p } ) = V ( p ). Now we can consider the two topologiesassociated with the spectral space X = Spec Z ( A ): the patch topology and theflat topology. We will denote Spec P ( A ) = X P and Spec F ( A ) = X F ; P = P A will be the family of open sets in Spec P ( A ) and F = F A the family of open setsin Spec F ( A ). Remark 5.3 (i) The family { D ( c ) T V ( d ) | c, d ∈ K ( A ) } is a basis of open setsfor Spec P ( A ) ;(ii) The family { V ( c ) | c ∈ K ( A ) } is a basis of open sets for Spec F ( A ) . Remark 5.4 (i) The patch topology on
Spec ( A ) is finer than the spectral andthe flat topologies on Spec ( A ) ( i.e. Z ⊆ P and
F ⊆ P );(ii) The inclusions
Z ⊆ P and
F ⊆ P show that the identity functions id : Spec P ( A ) → Spec Z ( A ) and id : Spec P ( A ) → Spec F ( A ) are continuous. Proposition 5.5
The two inverse functions u : Spec ( A ) → Spec Id ( L ( A )) and v : Spec Id ( L ( A )) → Spec ( A ) from Proposition 3.7 are homeomorphisms w.r.tthe patch and the flat topologies. Proof.
Applying Lemma 3.6 it is easy to prove that for all c ∈ K ( A ), thefollowing equalities u − ( V Id ( λ A ( c ))) = V ( c ) and u − ( D Id ( λ A ( c ))) = D ( c ) hold.Therefore, by Remarks 5.2 and 5.3, u is patch and flat continuous.For any p ∈ Spec ( A ), let us denote Λ( p ) = { q ∈ Spec ( A ) | q ≤ p } .10 roposition 5.6 For any p ∈ Spec ( A ) , the flat closure of the set { p } is cl F { p } = Λ( p ) . Proof.
According to the definition of the closure cl F ( p ) = cl F { p } , the followingequalities hold: cl F ( p ) = { q ∈ Spec ( A ) |∀ c ∈ K ( A )( q ∈ V ( c ) ⇒ V ( c ) T { p } 6 = ∅} )= { q ∈ Spec ( A ) |∀ c ∈ K ( A )( c ≤ q ⇒ c ≤ p ) } .In order to prove that cl F ( p ) ⊆ Λ( p ), let us consider q ∈ cl F ( p ) and c ∈ K ( A ). Then c ≤ q implies c ≤ q , therefore q = W { c ∈ K ( A ) | c ≤ q } ≤ W { c ∈ K ( A ) | c ≤ p } = p .Conversely, assume that q ∈ Λ( p ), so q ≤ p . Thus for any c ∈ K ( A ), c ≤ q implies c ≤ p , hence q ∈ cl F ( p ). Proposition 5.7 If S ⊆ Spec ( A ) is compact in Spec Z ( A ) then its flat closureis cl F ( S ) = [ p ∈ S Λ( p ) . Proof.
Applying Proposition 5.6, it follows that for any p ∈ S we have Λ( p )= cl F ( p ) ⊆ cl F ( S ), so [ p ∈ S Λ( p ) ⊆ cl F ( S ). Let us prove the converse inclusion cl F ( S ) ⊆ [ p ∈ S Λ( p ). Assume by absurdum that there exists q ∈ cl F ( S ) − [ p ∈ S Λ( p ),so q p for all p ∈ S . Then for all p ∈ S there exists c p ∈ K ( A ) such that c p p and c p ≤ q . This means that S ⊆ [ p ∈ S D ( c p ), so S ⊆ n [ i =1 D ( c p i ) for some p , . . . , p n ∈ S . Denote c = n _ i =1 c p i , so c ∈ K ( A ) and S ⊆ D ( c ). One remarksthat c ≤ q , so q ∈ V ( c ). Since q ∈ cl F ( S ) and V ( c ) is an open neighbourhoud of q in the flat topology, it follows that S T V ( c ) = ∅ . This contradicts S ⊆ D ( c ),hence cl F ( S ) ⊆ [ p ∈ S Λ( p ). We conclude that cl F ( S ) = [ p ∈ S Λ( p ).An element a ∈ A is regular if it is a join of complemented elements. Amaximal element in the set of proper regular elements is called max- regular.The set Sp ( A ) of max- regular elements of A is called the Pierce spectrum ofthe quantale A . For any proper regular element a there exists p ∈ Sp ( A ) suchthat a ≤ p . If e ∈ B ( A ) then we denote U ( e ) = { p ∈ Sp ( A ) | e a } . Thus it iseasy to prove that the family ( U ( e )) e ∈ B ( A ) is a basis of open sets for a topologyon Sp ( A ).For any p ∈ Spec ( A ) we define s A ( p ) = W { e ∈ B ( A ) | e ≤ p } ; s A ( p ) is regularand s A ( p ) ≤ p < Lemma 5.8 s A ( p ) is a max - regular element of A . roof. In order to prove that s A ( p ) ∈ Sp ( A ) is suffices to have: e ∈ B ( A ) and e p implies s A ( p ) ∨ e = 1. Assume by absurdum that there exists e ∈ B ( A )such that e p and s A ( p ) ∨ e <
1. The element s A ( p ) ∨ e is regular so thereexists a max - regular element q such that s A ( p ) ∨ e ≤ q . Since e p and p ∈ Spec ( A ) we have ¬ e ≤ p , so 1 = e ∨ ¬ ≤ p . This contradiction shows that s A ( p ) ∈ Sp ( A ).According to the previous lemma, for each p ∈ Spec ( A ), s A ( p ) is a max -regular element of A , so one obtains a function s A : Spec ( A ) → Sp ( A ). Proposition 5.9 Sp ( A ) is a Boolean space and s A : Spec ( A ) → Sp ( A ) issurjective and continuous w.r.t. both flat and spectral topologies on Spec ( A ) . Proof.
Assume that q ∈ Sp ( A ) then q ≤ p for some p ∈ Spec ( A ), hence q ≤ s A ( p ). Since q and s A ( p ) are max - regular we have q = s A ( p ), so s A is surjective.It is easy to see that Sp ( A ) is a Hausdorff space and ( U ( e )) e ∈ B ( A ) is basis ofclopen set for Sp ( A ). For all e ∈ B ( A ) we have s − A ( U ( e )) = D ( e ) = V ( ¬ e ),hence the functions s A : Spec Z ( A ) → Sp ( A ) and s A : Spec F ( A ) → Sp ( A ) arecontinuous. Therefore the topological space Sp ( A ) = Im ( s A ) is compact. Weconclude that Sp ( A ) is a Boolean space. The hyperarchimedean quantales were introduced in [9], where a characteri-zation theorem of these objects was proven. This section contains new algebraicand topological characterizations of the hyperarchimedean quantales. Let A be acoherent quantale. By [9], A is said to be hyperarchimedean if for any c ∈ K ( A )there exists an integer n > c n ∈ B ( A ). Applying Lemma 4.2, it fol-lows that a coherent frame A is hyperarchimedean if and only if B ( A ) = K ( A )(see [31]). Remark 6.1
Recall from [6] that a frame L is zero - dimensional if any element a ∈ A is a joint of complemented elements. On the other hand, by Remark2.1,(i) of [31], an algebraic frame is zero - dimensional if and only if eachcompact element is complemented. Thus a coherent frame is hyperarchimedeanif and only if it is zero - dimensional. Proposition 6.2 If A is a coherent quantale the the following are equivalent:(1) A is hyperarchimedean;(2) L ( A ) is a Boolean algebra;(3) Spec ( A ) = M ax ( A ) ;(4) The quantale [ ρ (0)) A is hyperarchimedean; R ( A ) is a hyperarchimedean frame;(6) R ( A ) is a zero - dimensional frame; Proof.
The equivalence of (1),(2),(3) and (6) was proven in [9], but for sake ofcompleteness we shall present a short proof of the proposition.(1) ⇔ (2) In accordance to Lemma 4.3, the following assertions are equiva-lent: • L ( A ) is a Boolean algebra: • for all c ∈ K ( A ), λ A ( c ) ∈ B ( L ( A )); • for all c ∈ K ( A ), there exists an integer n > c n ∈ B ( A ); • A is hyperarchimedean.(2) ⇔ (3) By the Nachbin theorem [5] and Proposition 3.7, L ( A ) is a Booleanalgebra iff Spec Id ( L ( A )) = M ax Id ( L ( A )) iff Spec ( A ) = M ax ( A ).(3) ⇔ (4) This equivalence follows by using Lemma 2.6.(1) ⇔ (5) According to Lemma 6 of [9], Spec ( A ) = Spec ( R ( A )) and M ax ( A ) = M ax ( R ( A )), therefore this equivalence follows by using that (1) and (3) areequivalent.(5) ⇔ (6) By Remark 6.1. Lemma 6.3 If L is a bounded distributive lattice then the following are equiv-alent:(1) L is a Boolean algebra;(2) Spec Id ( L ) = M ax Id ( L ) ;(3) For all distinct prime ideals M and N of L there exist x / ∈ M and y / ∈ N such that x ∧ y = 0 . Proof. (1) ⇔ (2) By the Nachbin theorem.(1) ⇒ (3) Assume that L is a Boolean algebra and M, N are distinct primeideals of L . Thus M, N are distinct maximal ideals of L so there exists anelement x ∈ M − N . Denoting y = ¬ x one gets x / ∈ N , y / ∈ M and x ∧ y = 0.(3) ⇒ (1) Assume by absurdum that there exist two ideals M and N of L such that M N . By hypothesis there exist two elements x and y of L suchthat x / ∈ M , y / ∈ N and x ∧ y = 0. Since x / ∈ M and x ∧ y = 0 implies y ∈ M , itfollows a contradiction, so M = N . It follows that Spec Id ( L ) = M ax Id ( L ). Proposition 6.4
Assume that A is a coherent quantale. The following follow-ing properties are equivalent:(1) For all distinct p, q ∈ Spec ( L ( A )) there exist c, d ∈ K ( A ) such that c p , d q and cd = 0 ;(2) For all distinct prime ideals I, J of L ( A ) there exist two elements x, y of L ( A ) such that x / ∈ I , y / ∈ J and x ∧ y = 0 . roof. Assume that
I, J are two distinct prime ideals of L ( A ). In accordanceto Proposition 3.7 there exist two p, q ∈ Spec ( A ) such that I = p ∗ , J = q ∗ and p = q . By hypothesis there exist c, d ∈ K ( A ) such that c p , d q and cd = 0.Applying Lemmas 3.2 and 3.6 one gets λ A ( c ) ∧ λ A ( d ) = λ A ( cd ) = λ A (0) = 0and λ A ( c ) / ∈ p ∗ , λ A ( d ) / ∈ q ∗ .Assume now that p, q ∈ Spec ( A ), with p = q so p ∗ , q ∗ are distinct primeideals of L ( A ). Thus there exist c, d ∈ K ( A ) such that λ A ( c ) / ∈ p ∗ , λ A ( d ) / ∈ q ∗ and λ A ( cd ) = λ A ( c ) ∧ λ A ( d ) = 0. Applying Lemma 3.6 one obtains c p and d q . Since A is semiprime, λ A ( cd ) = 0 implies cd = 0 (by Lemma 3.2,(9)),so there exist an integer n ≥ c n d n = 0. Let us denote u = c n and v = d n . Thus u and v are two compact elements of A such that u p , v q (because p, q are m - prime elements) and uv = 0. Proposition 6.5
Assume that A is a coherent quantale. The following proper-ties are equivalent:(1) A is hyperarchimedean;(2) L ( A ) is a Boolean algebra;(3) For all distinct prime ideals I, J of L ( A ) there exist two elements x, y of L ( A ) such that x / ∈ I , y / ∈ J and x ∧ y = 0 ;(4) For all distinct p, q ∈ Spec ( A ) there exist c, d ∈ K ( A ) such that c p , d q and cd = 0 ; Proof. (1) ⇔ (2) By Proposition 6.2.(2) ⇔ (3) By Lemma 6.3.(3) ⇔ (1) By Proposition 6.4. Corollary 6.6 If A is a coherent quantale then it is hyperarchimedean if andonly if for all distinct p, q ∈ Spec ( A ) there exist c, d ∈ K ( A ) such that c p , d q and cd = ρ (0) . Lemma 6.7 [43] Assume that X and Y are topological spaces, X is compactand Y is Hausdorff. Any continuous function f : X → Y is a closed map.Moreover, if f is bijective, then it is a homeomorphism. Theorem 6.8 If A is a coherent quantale then the following are equivalent:(1) A is hyperarchimedean;(2) For all distinct p, q ∈ Spec ( A ) there exist c, d ∈ K ( A ) such that c p , d q and cd = 0 ; Spec Z ( A ) is Hausdorff;(4) Spec Z ( A ) is a Boolean space;(5) Z = P ;(6) Spec F ( A ) is Hausdorff;(7) Spec F ( A ) is Boolean space;(8) Z = F . Proof.
The equivalence (1) ⇔ (2) follows from Proposition 6.5 and the equivalences(3) ⇔ (4), (6) ⇔ (7) are well - known from the general topology.(2) ⇒ (3) Let p, q be two distinct elements of Spec ( A )). Then there exist c, d ∈ K ( A ) such that c p , d q and cd = 0. It follows that p ∈ D ( c ), q ∈ D ( d ) and D ( c ) T D ( d ) = D ( cd ) = ∅ . Thus Spec Z ( A ) is a Hausdorff space.(3) ⇒ (5) Assume that Spec Z ( A ) is a Hausdorff space. Recall from Remark5.4,(ii) that the identity function id : Spec P ( A ) → Spec Z ( A ) is continuous.Since Spec P ( A ) is compact (cf. Lemma 5.1) and Spec Z ( A ) is Hausdorff, byLemma 6.7 it follows that the map id : Spec P ( A ) → Spec Z ( A ) is a homeomor-phism, hence Z = P .(5) ⇒ (1) Assume that Z = P and p ∈ Spec ( A ). We want to show that p ∈ M ax ( A ). According to Remark 5.4(i) and the hypothesis (5) we have F ⊆ P = Z . From F ⊆ P it follows that any closed set in
Spec F ( A ) isclosed in Spec Z ( A ), so cl Z ( { p } ) ⊆ cl P ( { p } ). Thus by applying Proposition5.6 one gets V ( p ) ⊆ Λ( p ). Thus V ( p ) = { p } , so p ∈ M ax ( A ). It followsthat Spec ( A ) = M ax ( A ), hence, by Proposition 6.2 we conclude that A ishyperarchimedean.(1) ⇒ (6) Assume that A is hyperarchimedean and p, q are distinct elementsof Spec ( A ), hence, by Proposition 6.2 we have p, q ∈ M ax ( A ), therefore p ∨ q = 1.Since 1 ∈ K ( A ) there exist p, q ∈ K ( A ) such that c ≤ p , d ≤ q and c ∨ d = 1.Then p ∈ V ( c ), q ∈ V ( d ) and V ( c ) , V ( d ) are open sets of Spec F ( A ) such that V ( c ) T V ( d ) = V ( c ∨ d ) = V (1) = ∅ . It follows that Spec F ( A ) is a Hausdorffspace.(6) ⇒ (1) Assume that Spec F ( A ) is a Hausdorff space. Let p ∈ Spec ( A ) and q ∈ M ax ( A ) such that p ≤ q . Since Spec F ( A ) is Hausdorff we have cl F ( { q } ) = { q } . According to Proposition 5.6 we have p ∈ Λ( q ) = cl F ( { q } ) = { q } , hence p = q . Thus Spec ( A ) = M ax ( A ), so A is hyperarchimedean.(6) ⇒ (8) Assume that Spec F ( A ) is a Hausdorff space. The identity function id : Spec P ( A ) → Spec F ( A ) is continuous, Spec F ( A ) is compact and Spec F ( A ) isHausdorff. Hence by Lemma 6.7 it follows that id : Spec P ( A ) → Spec F ( A ) is ahomeomorphism, so F = P . According to the previous proof of the implications(6) ⇒ (1), and (1) ⇒ (5) we have Z = P , therefore Z = F .153) ⇒ (5) Assume that Z = F . If p ∈ Spec ( A ) then Λ( p ) = cl F ( { p } ) = { p } = V ( p ), hence p ∈ M ax ( A ). Thus Spec ( A ) = M ax ( A ), so the quantale A ishyperarchimedean. Remark 6.9
If we apply the previous theorem to the quantale Id ( R ) of theideals of a commutative ring R then we obtain the main part of Theorem 3.3from [1]. In this section we shall study the flat topology on the maximal spectrum ofcoherent quantales in order to obtain new results on the normal and B - normalquantales.We fix a coherent quantale A . Recall that M ax F ( A ) is the maximal spec-trum M ax ( A ) of A endowed with the restriction of the flat topology of Spec ( A ). Lemma 7.1 If c ∈ K ( A ) then V ( c ) T M ax ( A ) is a clopen set of M ax F ( A ) . Proof.
Assume that c is a compact element of A . According to Remark 5.3,(ii), V ( c ) T M ax ( A ) is an open set of M ax F ( A ). It remains to prove that the set D ( c ) T M ax ( A ) is open in M ax F ( A ). Let p ∈ D ( c ) T M ax ( A ), hence c p and p ∈ M ax ( A ). If c ∨ p < p < c ∨ p ≤ q for some q ∈ M ax ( A ),contradicting the maximality of p . Thus c ∨ p = 1, hence there exists d ∈ K ( A )such that d ≤ p and c ∨ d = 1. One obtains V ( c ) T V ( d ) = V ( c ∨ d ) = V (1) = ∅ , hence V ( d ) ⊂ D ( c ). It follows that p ∈ V ( d ) T M ax ( A ) ⊆ D ( c ) T M ax ( A ),so D ( c ) T M ax ( A ) is an open subset of M ax F ( A ). Proposition 7.2
The topological space
M ax F ( A ) is Hausdorff and zero - di-mensional. Proof.
Let p, q be two distinct maximal elements of A , hence p ∨ q = 1.Thus there exist c, d ∈ K ( A ) such that c ≤ p , d ≤ q and c ∨ d = 1, there-fore p ∈ V ( c ), d ∈ V ( d ) and V ( c ) T V ( d ) = V ( c ∨ d ) = V (1) = ∅ . It resultsthat M ax F ( A ) is a Hausdorff space. In accordance to Lemma 7.1, the family( V ( c ) T M ax ( A )) c ∈ K ( A ) is a basis of clopen sets for M ax F ( A ), so this topolog-ical space is zero - dimensional.Following [9] we shall denote r ( A ) = V M ax ( A ) . One remarks that r ( A )extends the notion of Jacobson radical of a commutative ring. It is obvious that ρ (0) ≤ r ( A ). Theorem 7.3
M ax F ( A ) is compact if and only if [ r ( A )) A is a hyperarchimedeanquantale. roof. ( ⇒ ) Assume that M ax F ( A ) is compact. First we shall prove that L ( A ) / ( r ( A )) ∗ is a Boolean algebra. Let c be a compact element of A such that λ A ( c ) / ( r ( A )) ∗ =0 / ( r ( A )) ∗ , so λ A ( c ) ( r ( A )) ∗ . By Lemma 3.5 we have c r ( A ), so there exists m c ∈ M ax ( A ) such that c m c . For any m ∈ M ax ( A ) we have c ≤ m or c m ; if c m then there exists d m ∈ K ( A ) such that d m ≤ m and c ∨ d m = 1.Since c m c , the family { m ∈ M ax ( A ) | c m } is non-empty. One remarks that M ax ( A ) = { m ∈ M ax ( A ) | c ≤ m }∪{ m ∈ M ax ( A ) | c m } ⊆ V ( c ) ∪ [ c m V ( d m ).By hypothesis M ax F ( A ) is compact so there exist m , . . . , m n ∈ M ax ( A )such that c m i for i = 1 , . . . , n and M ax ( A ) ⊆ V ( c ) ∪ n [ i =1 V ( d m i ). Let usdenote d i = d m i for i = 1 , . . . , n and d = d d . . . d m . Therefore M ax ( A ) ⊆ V ( c ) ∪ V ( d ) = V ( c ), so cd ≤ r ( A ). By Lemma 3.5, cd ≤ r ( A ) implies λ A ( cd ) ∈ ( r ( A )) ∗ .According to Lemma 2.1 (i), from c ∨ d i = 1, i = 1 , . . . , n one gets c ∨ d = 1.Since λ A ( c ) ∨ λ A ( d ) = λ A ( c ∨ d ) = λ A (1) = 1 and λ A ( c ) ∧ λ A ( d ) = λ A ( cd ), thefollowing equalities hold: λ A ( c ) / ( r ( A )) ∗ ∨ λ A ( d ) / ( r ( A )) ∗ = 1 / ( r ( A )) ∗ ; λ A ( c ) / ( r ( A )) ∗ ∧ λ A ( d ) / ( r ( A )) ∗ = λ A ( cd ) / ( r ( A )) ∗ = 0 / ( r ( A )) ∗ .It follows that L ( A ) / ( r ( A )) ∗ is a Boolean algebra. By Proposition 6 of [9],the lattices L ([ r ( A )) A ) and L ( A ) / ( r ( A )) ∗ are isomorphic, so the reticulation L ([ r ( A )) A ) of the quantale [ r ( A ) A is a Boolean algebra. Applying Proposition6.1, it follows that [ r ( A )) A is a hyperarchimedian quantale.( ⇐ ) Assume that the quantale [ r ( A )) A is hyperarchimedean. By Proposition6.2 we have M ax F ([ r ( A )) A ) = Spec F ([ r ( A )) A ), so M ax F ([ r ( A )) A ) is compact(cf. Lemma 5.1). It is easy to see that M ax F ( A ) = M F ([ r ( A )) A ), so M ax F ( A )is compact. Proposition 7.4
The topology of
M ax F ( A ) is finer than the topology of M ax Z ( A ) . Proof.
A basic open subset of
M ax Z ( A ) has the form U = M ax ( A ) T D ( c ),for some c ∈ K ( A ). Let us consider an element m ∈ U so m ∈ M ax ( A ) and c m , hence c ∨ m = 1. Thus we have c ∨ d = 1 for some d ∈ K ( A ) with d ≤ m . Therefore V ( c ) T V ( d ) = V ( c ∨ d ) = V (1) = ∅ , so m ∈ M ax ( A ) T V ( d )and M ax ( A ) T V ( d ) is included in U . It follows that U is an open subset of M ax F ( A ).The following proposition characterizes the quantales A for which M ax Z ( A )and M ax F ( A ) coincide. Proposition 7.5 If A is a coherent quantale then the following are equivalent:(1) M ax F ( A ) is compact;
2) The topological spaces
M ax Z ( A ) and M ax F ( A ) coincide;(3) [ r ( A )) A is a hyperarchimedean quantale. Proof.
The equivalence of (1) and (3) follows by Theorem 7.3. We remark thatthe following equalities hold:
M ax F ( A ) = M ax F ([ r ( A )) A ) and M ax Z ([ r ( A )) A )= M ax Z ( A ). According to Theorem 6.8 the assertions (2) and (3) are equiva-lent.Following [25], p.199 we say that a commutative ring R is a Gelfand ringif each prime ideal of R is contained in a unique maximal ideal. Recall from[40],[25] that a bounded distributive lattice L is called normal if for all elements x, y ∈ L such that x ∨ y = 1 there exist u, v ∈ L such that x ∨ u = y ∨ v = 1and uv = 0. We know from [25], p.68 that a bounded distributive lattice L isnormal if and only if each prime ideal of L is contained in a unique maximalideal. The normal quantales were introduced in [35] as an abstractization of thelattices of ideals of Gelfand rings and normal lattices.According to [35], a quantale A is said to be normal if for all a, b ∈ A suchthat a ∨ b = 1 there exist e, f ∈ A such that a ∨ e = b ∨ f = 1 and ef = 0. If1 ∈ K ( A ) then A is normal if and only if for all c, d ∈ K ( A ) such that c ∨ d = 1there exist e, f ∈ K ( A ) such that c ∨ e = d ∨ f = 1 and ef = 0 (cf. Lemma 20of [9]). One observes that a commutative ring R is a Gelfand ring iff Id ( R ) isa normal quantale and a bounded distributive lattice L is normal iff Id ( L ) is anormal frame.The normal quantales offer an abstract framework in order to unify somealgebraic and topological properties of commutative Gelfand rings [25], [23],[32], [29], [39], normal lattices [25], [19], [37], [40], commutative unital l - groups[7], F - rings [7], [25], M V - algebras and BL - algebras [16], [28], Gelfandresiduated lattices [18], etc.Let us fix a coherent quantale A . Proposition 7.6 [9] The quantale A is normal if and only if the reticulation L ( A ) is a normal lattice ( in the sense of [40],[25]). Proposition 7.7 [35],[20],[41] If A is a coherent quantale then the followingare equivalent:(1) A is a normal quantale;(2) For all distinct m, n ∈ M ax ( A ) there exist c , c ∈ K ( A ) such that c m , c n and c c = 0 ;(3) The inclusion M ax ( A ) ⊆ Spec ( A ) is a Hausdorff embedding (i.e. anydistinct points in M ax ( A ) have disjoint neighbourhoods in Spec Z ( A )) ;(4) For any p ∈ Spec ( A ) there exists a unique m ∈ M ax ( A ) such that p ≤ m ;(5) Spec Z ( A ) is a normal space;
6) The inclusion
M ax Z ( A ) ⊆ Spec Z ( A ) has a continuous retraction γ : Spec Z ( A ) → M ax Z ( A ) ;(7) If m ∈ M ax ( A ) then Λ( m ) is a closed subset of Spec Z ( A ) . Remark 7.8
A proof of the previous proposition can be obtained by using Propo-sition 7.6 and some characterizations of normal lattices given in [19], [25], [37],[40].
Theorem 7.9
Assume that A is a normal quantale. Then the retraction map γ : Spec ( A ) → M ax ( A ) is flat continuous if and only if M ax F ( A ) is a compactspace. Proof.
Assume that the retraction map γ : Spec F ( A ) → M ax F ( A ) is continu-ous. Since Spec F ( A ) is compact it follows that M ax F ( A ) is also compact.Conversely, assume that M ax F ( A ) is compact, hence by Proposition 7.2it is a Boolean space. By Proposition 7.3 it results that [( r ( A ) A ) is a hy-perarchimedean quantale. Applying the condition (4) of Theorem 6.7 one gets M ax Z ([( r ( A ) A ) = M ax F ([( r ( A )) A ). We remark that M ax Z ( A ) and M ax F ([( r ( A )) A )are homeomorphic, thus by Theorem 6.7(4) it follows that M ax Z ( A ) is a Booleanspace. Thus ( M ax ( A ) T D ( c )) c ∈ K ( A ) is a basis of clopen sets in M ax Z ( A ).Let us consider an element c ∈ K ( A ); in accordance to the continuity of γ : Spec F ( A ) → M ax F ( A ), it follows that γ − ( M ax ( A ) T D ( c )) is a clopen setin Spec Z ( A ). Applying Lemma 2.4 of [9] we find an element e ∈ B ( A ) suchthat γ − ( M ax ( A ) T D ( c )) = V ( e ). Therefore γ − ( M ax ( A ) T D ( c )) is a clopensubset of M ax F ( A ) (cf. Lemma 7.1), hence the map γ : Spec F ( A ) → M ax F ( A )is continuous. Proposition 7.10 If M ax Z ( A ) is Hausdorff and ρ (0) = r ( A ) then the quantale A is normal. Proof.
Assume by absurdum that the quantale A is not normal, so by Propo-sition 7.7,(4) there exist p ∈ Spec ( A ) and q, r ∈ M ax ( A ) such that q = r , p ≤ q and p ≤ r . Since M ax Z ( A ) is Hausdorff there exist c, d ∈ K ( A ) such that q ∈ D ( c ), r ∈ D ( d ) and D ( cd ) T M ax ( A ) = D ( c ) T D ( d ) T M ax ( A ) = ∅ .If cd ρ (0) then cd r ( A ), so cd m for some m ∈ M ax ( A ). It results that m ∈ D ( cd ) T M ax ( A ), contradicting D ( cd ) T M ax ( A ) = ∅ . Thus cd ≤ ρ (0),hence one gets c ≤ p or d ≤ p . If c ≤ p then c ≤ q , contradicting q ∈ D ( c );similarly, c ≤ p contradicts r ∈ D ( d ). We conclude that A is normal. Corollary 7.11
M ax Z ( A ) is a Hausdorff space if and only if [ r ( a )) A is a nor-mal quantale. roof. We observe that the quantale C = [ r ( a )) A verifies the conditions ρ C (0)= r ( C ) and M ax Z ( A ) = M ax C ( A ). Applying Proposition 7.10 to the quan-tale C the following equivalences hold: M ax Z ( A ) is Hausdorff iff M ax Z ( C ) isHausdorff iff C is a normal quantale.Following [9] we say that a quantale A is said to be B - normal if for all c, d ∈ K ( A ) there exist e, f ∈ B ( A ) such that c ∨ e = d ∨ f = 1 and cd = 0. Ifthe B - normal quantale A is a frame then we shall say that A is a B - normalframe. The B - normal quantales constitute an abstract setting in which wecan generalize various results on the clean commutative rings [23], [34], the B -normal (bounded distributive ) lattices [10], the clean unital l - groups [21], thequasi - local BL - algebras [28], the quasi - local residuated lattices [33],etc. Lemma 7.12 If A is normal quantale then ( D ( e ) T M ax ( A )) e ∈ B ( A ) is the fam-ily of the clopen subsets of M ax Z ( A ) . Proof.
Let K be a clopen subset of M ax Z ( A ). If γ : Spec Z ( A ) → M ax Z ( A ) isthe continuous retract of the inclusion M ax Z ( A ) ⊆ Spec Z ( A ), then L = γ − ( K )is a clopen set in Spec Z ( A ). By Lemma 24 of [9] there exists an element e ∈ B ( A ) such that L = D ( e ). Thus K = γ ( D ( e )) = { γ ( p ) | p ∈ Spec ( A ) , e p } . Itis easy to see that for all p ∈ Spec ( A ) we have e ≤ p iff e ≤ γ ( p ), hence K = { γ ( p ) | p ∈ Spec ( A ) , e γ ( p ) } = D ( e ) T M ax ( A ).The following theorem contains some conditions that characterize the B -normal algebras. Theorem 7.13 If A is a coherent quantale then the following are equivalent:(1) A is B - normal;(2) R ( A ) is a B - normal frame;(3) The reticulation L ( A ) is a B - normal lattice;(4) For all distinct p, q ∈ M ax ( A ) there exists e ∈ B ( A ) such that e ≤ p and ¬ e ≤ q ;(5) A is a normal quantale and M ax Z ( A ) is a zero - dimensional space;(6) A is a normal quantale and M ax Z ( A ) is a Boolean space;(7) The family ( D ( e ) T M ax ( A )) e ∈ B ( A ) is a basis of open sets for M ax Z ( A ) ;(8) The function s A | Max ( A ) : M ax Z ( A ) → Sp ( A ) is a homeomorphism. Proof.
The equivalence of the properties (1) , (2) , (3) , (5) and (6) was establishedin [9], hence it remains to prove the equivalence of the other conditions.(1) ⇒ (4) Let p, q be two distinct maximal elements of A , hence p ∨ q = 1.Since A is B - normal, there exist e, f ∈ B ( A ) such that p ∨ f = q ∨ e = 1 and20 f = 0. From p ∨ f = q ∨ e = 1 one gets f p , e q , hence ¬ f ≤ p and ¬ e ≤ q .The equality ef = 0 implies e ≤ ¬ f ≤ p .(6) ⇒ (7) Since M ax Z ( A ) is a Boolean space, the family of its clopen subsetsis a basis of open sets. By Lemma 7.12, the family ( D ( e ) T M ax ( A )) e ∈ B ( A ) isexactly this basis of open sets for M ax Z ( A ).(7) ⇒ (4) Let p, q be two distinct maximal elements of A . We observe that U = Spec ( A ) −{ q } = Spec ( A ) − V ( q ) = D ( q ) is open in Spec Z ( A ), so U T M ax ( A )is an open subset of M ax Z ( A ) that contains p . In accordance to the hypothesis,there exists e ∈ B ( A ) such that p ∈ D ( e ) T M ax ( A ) ⊆ U T M ax ( A ). It followsthat e p and e ≤ q , so ¬ e ≤ p and e ≤ q .(1) ⇒ (8) Assume that A is B - normal. In accordance to Proposition 5.9, s A | Max ( A ) : Spec Z ( A ) → Sp ( A ) is a surjective continuous map. We shall provethat the restriction of s A to M ax ( A ) is injective. Let m, n ∈ M ax ( A ) such that m = n . We know that the conditions (1) and (4) are equivalent, so there exists e ∈ B ( A ) such that e ≤ m , ¬ e ≤ n , hence e ≤ s A ( m ) and e s A ( n ). It followsthat s A ( m ) = s A ( n ), so s A is injective.In order to prove that s A | Max ( A ) : M ax Z ( A ) → Sp ( A ) is surjective assumethat q ∈ Sp ( A ), hence q = s A ( p ), for some p ∈ Spec ( A ). Let γ ( p ) be the uniquemaximal element of A such that p ≤ γ ( p ). Thus q = s A ( p ) ≤ s A ( γ ( p )), so q = γ ( p ), because q and γ ( p ) are max- regular elements. We know already that(1) and (6) are equivalent, so M ax ( A ) is a Boolean space. By Proposition 5.9, Sp ( A ) is also a Boolean space. Therefore, by applying Lemma 6.8 it followsthat s A | Max ( A ) : M ax Z ( A ) → Sp ( A ) is a homeomorphism.(8) ⇒ (6) Taking into account the hypothesis (8), it follows that the function( s A | Max ( A ) ) − ◦ s A : Spec Z ( A ) → M ax Z ( A ) is a continuous retraction of theinclusion M ax Z ( A ) ⊂ Spec Z ( A ), so A is a normal quantale. Moreover, by (8)and Proposition 5.9 it follows that M ax Z ( A ) is a Boolean space. Corollary 7.14
Let A be a normal quantale. If [ r ( A )) A is a hyperarchimedeanquantale then A is B - normal. Proof.
Assume that [ r ( A )) A is hyperarchimedean, so by Proposition 4.5 wehave M ax F ( A ) = M ax Z ( A ). Therefore by using Proposition 7.2 it follows that M ax Z ( A ) is zero - dimensional. Applying Theorem 7.13,(5) one gets that A isa B - normal quantale. If A is a quantale then we denote by M in ( A ) the set of minimal m - primeelements of A ; M in ( A ) is called the minimal prime spectrum of A . If 1 ∈ K ( A )then for any p ∈ Spec ( A ) there exists q ∈ M in ( A ) such that q ≤ p . For anybounded distributive lattice L we denote by M in Id ( L ) the set of minimal primeideals in L ; M in Id ( L ) is the minimal prime spectrum of the frame Id ( L ).21e will obtain a description of the minimal m - prime elements of a coherentquantale A by using the reticulation. First we remember from [40] the followingresult. Lemma 8.1
A prime ideal P of a bounded distributive lattice L is minimalprime if and only if for all x ∈ P we have Ann ( x ) P . Let us fix a coherent quantale A . Lemma 8.2 If c ∈ K ( A ) and p ∈ Spec ( A ) then Ann ( λ A ( c )) ⊆ p ∗ if and onlyif c → ρ (0) ≤ p . Proof. If Ann ( λ A ( c )) ⊆ p ∗ , then by using Lemma 3.4 and Proposition 4.5,one gets c → ρ (0) ≤ ρ ( c → ρ (0)) = (( c → ρ (0)) ∗ ) ∗ = ( Ann ( λ A ( c ))) ∗ ≤ ( p ∗ ) ∗ = p . Conversely, if c → ρ (0) ≤ p , then by using Proposition 4.5 we have Ann ( λ A ( c )) = ( c → ρ (0)) ∗ ⊆ p ∗ . Proposition 8.3 If p ∈ Spec ( A ) then the following are equivalent:(1) p ∈ M in ( A ) ;(2) p ∗ ∈ M in Id ( L ( A )) ;(3) For all c ∈ K ( A )) , λ A ( p ) ∈ p ∗ implies Ann ( λ A ( p )) p ∗ ;(4) For all c ∈ K ( A )) , c ≤ p if and only if c → ρ (0) p . Proof. (1) ⇔ (2) Let us consider the order - preserving map u : Spec ( A ) → Spec Id ( L ( A )) defined by u ( p ) = p ∗ , for all p ∈ Spec ( A ). According to Propo-sition 3.7, u is an order - isomorphism, hence the conditions (1) and (2) areequivalent.(2) ⇔ (3) By Lemma 3.3,(3), p ∗ is a prime ideal of the lattice L ( A ). There-fore, by using Lemma 8.1 it follows that the properties (2) and (3) are equivalent.(3) ⇔ (4) By Lemmas 3.6 and 8.2. Corollary 8.4 If A is semiprime and p ∈ Spec ( A ) then p ∈ M in ( A ) if andonly if for all c ∈ K ( A ) , c ≤ p implies c ⊥ p . Let us denote by
M in Z ( A ) (resp. M in F ( A )) the topological space obtainedby restricting the topology of Spec Z ( A ) (resp. Spec F ( A )) to M in ( A ). Sim-ilarly, for a bounded distributive lattice L we denote by M in
Id,Z ( L ) (resp. M in
Id,F ( L )) the space obtained by restricting to M in Id ( L ) the Stone topology(resp. the flat topology) of Spec Id ( L ). Lemma 8.5
The topological spaces
M in Z ( A ) and M in
Id,Z ( L ) (resp. M in F ( A ) and M in
Id,F ( L ) )) are homeomorphic. orollary 8.6 M in Z ( A ) is a zero - dimensional Hausdorff space and M in F ( A ) is a compact T space. Proof.
By [42],
M in
Id,Z ( L ) is a zero - dimensional Hausdorff space and M in
Id,F ( L ) is a compact T Proposition 8.7 [42] If L is a bounded distributive lattice then the followingare equivalent:(1) M in
Id,Z ( L ) = M in
Id,F ( L ) ;(2) M in
Id,Z ( L ) is a compact space;(3) M in
Id,Z ( L ) is a Boolean space;(4) For any s ∈ L there exists y ∈ L such that x ∧ y = 0 and Ann ( x ∨ y ) = { } . Theorem 8.8 If A is a semiprime quantale then the following are equivalent:(1) M in Z ( A ) = M in F ( A ) ;(2) M in Z ( A ) is a compact space;(3) M in Z ( A ) is a Boolean space;(4) For any c ∈ K ( A ) there exists d ∈ K ( A ) such that cd = 0 and ( c ∨ d ) ⊥ = . Proof. (1) ⇔ (2) ⇔ (3) These equivalences follow by applying Lemma 8.5 and tak-ing in account the equivalence of the assertions (i), (ii) and (iii) from Proposition8.7.(1) ⇒ (2) Assume that c ∈ K ( A ). By Lemma 8.5 we have M in
Id,Z ( L ( A )) = M in
Id,F ( L ( A )), therefore by applying Proposition 8.7 to the lattice L ( A ) thereexists d ∈ K ( A ) such that λ A ( cd ) = λ A ( c ) ∧ λ A ( d ) = 0 and Ann ( λ A ( c ∨ d ))= Ann ( λ A ( c ) ∨ λ A ( d )) = { } . The quantale A is semiprime, hence by usingLemma 3.2,(9) and Proposition 4.5, one obtains cd = 0 and (( c ∨ d ) ⊥ ) ∗ = { } .If z ∈ K ( A ) and z ≤ ( c ∨ d ) ⊥ then λ A ( z ) ∈ (( c ∨ d ) ⊥ ) ∗ , so λ A ( z ) = 0. Since A issemiprime it follows that z = 0 (cf. Lemma 3.2,(9)). We conclude that ( c ∨ d ) ⊥ = 0.(4) ⇒ (1) Assume that x ∈ L ( A ) hence x = λ A ( c ) for some c ∈ K ( A ). Thenthere exists d ∈ K ( A ) such that cd = 0 and ( c ∨ d ) ⊥ = 0. Denoting y = λ A ( d ) weobtain x ∧ y = λ A ( cd ) = 0 and Ann ( x ∨ y ) = Ann ( λ A ( c ∨ d )) = (( c ∨ d ) ⊥ ) ∗ = 0.By Proposition 8.7 we have M in
Id,Z ( L ( A )) = M in
Id,F ( L ( A )), hence M in Z ( A )= M in F ( A ). 23ecall from [1] that an mp - ring is a commutative ring R with the propertythat each prime ideal of R contains a unique minimal prime ideal. Let us extendthis notion to quantales: a quantale A is an mp - quantale if for any p ∈ Spec ( A )there exist a unique q ∈ M in ( A ) such that q ≤ p . An mp - frame is an mp -quantale wich is a frame. We remark that a ring R is an mp - ring if and onlyif the quantale Id ( R ) of ideals of R is an mp - quantale.The mp - quantales can be related to the conormal lattices, introduced byCornish in [11] under the name of ”normal lattices”. According to [40],[25], aconormal lattice is a bounded distributive lattice L such that for all x, y ∈ L with x ∧ y = 0 there exist u, v ∈ L having the properties x ∧ u = y ∧ v = 0 and u ∨ v = 1. In [11] Cornish obtained several characterizations of the conormallattices. Proposition 8.9 [11] A bounded distributive lattice L is conormal if and onlyif any prime ideal of L contains a unique minimal prime ideal. Corollary 8.10
A coherent quantale A is an mp - quantale if and only if thereticulation L ( A ) is a conormal lattice. Proof.
Recall that the two functions u A : Spec ( A ) → Spec Id ( L ( A )) and v A : Spec Id ( L ( A )) → Spec ( A ) from Proposition 3.7 are order - isomorphisms (theorder is the inclusion). Thus the corollary follows by using Proposition 8.9.By [22] for any bounded distributive lattice L there exists a commutativering R such that the lattices L and L ( A ) are isomorphic (see also the discussionfrom Section 3.13 of [25]). Thus for any coherent quantale A there exists acommutative ring R such that the lattices L ( A ) and L ( R ) are isomorphic (weshall identify these isomorphic lattices). Let us fix this ring R associated withthe quantale A .In accordance with Proposition 3.7 we have the following homeomorphisms: Spec Z ( A ) u A −−→ Spec
Id,Z ( L ( A )) v R −−→ Spec Z ( R ) (i) Spec Z ( R ) u R −−→ Spec
Id,Z ( L ( A )) v A −−→ Spec Z ( A ) (ii)By restricting these four maps to minimal prime spectra one gets the follow-ing homeomorphisms: M in Z ( A ) u A −−→ M in
Id,Z ( L ( A )) v R −−→ M in Z ( R ) (iii) M in Z ( R ) u R −−→ M in
Id,Z ( L ( A )) v A −−→ Spec Z ( A ) (iv)(we denote the restrictions by the same symbols). Remark 8.11
Taking into account Proposition 5.5 it is easy to prove that thefollowing maps:
Spec F ( A ) u A −−→ Spec
Id,F ( L ( A )) , Spec
Id,F ( L ( A )) v A −−→ Spec F ( A ) , Spec F ( R ) u R −−→ Spec
Id,F ( L ( A )) and Spec
Id,F ( L ( A )) v R −−→ Spec F ( R ) are homeo-morphisms. Lemma 8.12
The coherent quantale A is an mp - quantale if and only if R isan mp - ring. roof. We apply Proposition 8.9 to the isomorphic reticulations L ( A ) and L ( R ) of the quantale A and the ring R . Proposition 8.13
For a coherent quantale A the following are equivalent:(1) The inclusion M in F ( A ) ⊆ Spec F ( A ) has a flat continuous retraction;(2) The inclusion M in F ( R ) ⊆ Spec F ( R ) has a flat continuous retraction. Proof.
According to Remark 8.11, each of these two conditions is equivalentto the following property: the inclusion
M in
Id,F ( L ( A )) ⊆ Spec
Id,F ( L ( A )) hasa flat continuous retraction. Theorem 8.14 If A is a coherent quantale then the following are equivalent:(1) A is an mp - quantale;(2) For any distinct elements p, q ∈ M in ( A ) we have p ∨ q = 1 ;(3) R ( A ) is an mp - frame;(4) [ ρ (0)) A is an mp - quantale;(5) The inclusion M in F ( A ) ⊆ Spec F ( A ) has a flat continuous retraction;(6) Spec F ( A ) is a normal space;(7) If p ∈ M in ( A ) then V ( p ) is a closed subset of Spec F ( A ) . Proof. (1) ⇒ (2) Suppose that p, q are distinct elements of M in ( A ). If p ∨ q < m then p ∨ q ≤ m for some m ∈ M ax ( A ). Then there exist two distinct p, q ∈ M in ( A ) such that p ≤ m and q ≤ m , contradicting that A is an mp - quantale.It follows that p ∨ q = 1.(2) ⇒ (1) Assume by absurdum that there exist p ∈ Spec ( A ) and twodistinct q, r ∈ M in ( A ) such that q ≤ p and r ≤ p . Thus q ∨ r = 1, hence p = 1,contradicting that p ∈ Spec ( A ).(1) ⇔ (3) By Lemma 6 of [9] we have Spec ( A ) = Spec ( R ( A )), hence M in ( A ) = M in ( R ( A )), so the equivalence of (1) and (3) is immediate.(1) ⇔ (4) This equivalence follows from Spec ( A ) = Spec ([ ρ (0)) A ) and M in ( A ) = M in ([ ρ (0)) A ).(1) ⇔ (5) By using Lemma 8.12, Proposition 8.13 and the equivalence of theconditions (i), (v) from Theorem 6.2 of [1] it results that the properties (1) and(5) are equivalent.(1) ⇔ (6) By Remark 8.11, Lemma 8.12 and Theorem 6.2 of [1], it followsthat A is an mp - quantale iff R is an mp - ring iff Spec F ( R ) is normal iff Spec F ( A ) is normal. 252) ⇒ (7) Assume that p ∈ M in ( A ) and q ∈ D ( p ). Consider an element r ∈ M in ( A ) such that r ≤ q . Since p q we have p = r , hence p ∨ r = 1by the hypothesis (2), so there exist c, d ∈ K ( A ) such that c ≤ p , d ≤ r and c ∨ d = 1. Thus V ( c ) T V ( d ) = V ( c ∨ d ) = V (1) = ∅ , so V ( d ) ⊆ D ( c ) ⊆ D ( p ).From d ≤ r ≤ q one gets q ∈ V ( d ). Since q ∈ V ( d ) ⊆ D ( p ) and V ( d ) is a basicopen set of Spec F ( A ) it follows that D ( p ) is open in Spec F ( A ). We concludethat V ( p ) is closed in Spec F ( A ).(7) ⇒ (2) Assume by absurdum that there exist two distinct p, q ∈ M in ( A )such that p ∨ q <
1, so p ∨ q ≤ m for some m ∈ M ax ( A ). Therefore m ∈ V ( p )and m ∈ V ( q ), hence V ( p ) T V ( q ) = ∅ . Since V ( q ) is an open neighborhood of p in Spec F ( A ) and V ( p ) is flat closed, one gets q ∈ V ( p ). Thus q ≤ p so q = p because p and q are minimal m - prime elements. This contradiction shows thatfor all distinct minimal m - prime elements p, q we have p ∨ q = 1. Remark 8.15 If R is a commutative ring and A is the quantale Id ( R ) of idealsof R , then applying the previous result one obtains some of the characterizationsof mp - rings, contained in Theorem 6.2 of [1]. Recall from [4] that a commutative ring R is said to be an P F - ring if theannihilator of each element of R is a pure ideal. We shall generalize this notionto quantales. Then a quantale A is a P F - quantale if for each c ∈ K ( A ), c ⊥ isa pure element. For any commutative ring R , Id ( R ) is a P F - quantale if andonly if R is a P F - ring.
Proposition 8.16 [3] If A is a bounded distributive lattice L then the followingare equivalent(1) L is conormal;(2) For all x ∈ L , Ann ( x ) is a σ - ideal;(3) Any minimal prime ideal of L is a σ - ideal. In other words, a bounded distributive lattice L is conormal if and only if Id ( L ) is a P F - frame.In what follows we shall establish a relationship between
P F - quantales and mp - quantales. We fix a coherent quantale A . Lemma 8.17
Any
P F - quantale A is semiprime. Proof.
Let c be a compact element of A such that c n = 0 for some integer n ≥
1. Then c n − ≤ ( c n − ) ⊥ , hence ( c n − ) ⊥ = ( c n − ) ⊥ ∨ ( c n − ) ⊥ = 1, because( c n − ) ⊥ is pure. Thus c n − ≤ ( c n − ) ⊥ = 0 , so c n − = 0. By using many timesthis argument one gets c = 0. According to Lemma 2.4,(2) it follows that A issemiprime. 26 roposition 8.18 If A is a P F - quantale then the reticulation L ( A ) is aconormal lattice. Proof.
By Lemma 8.17, the
P F - quantale A is semiprime. Assume that x ∈ L ( A ) so x = λ A ( c ), for some c ∈ K ( A ). Applying Proposition 4.5 oneobtains Ann ( x ) = Ann ( λ A ( c )) = Ann ( c ∗ ) = ( c ⊥ ) ∗ . By hypothesis, c ⊥ is a pureelement of A , therefore by Lemma 4.7 it follows that Ann ( x ) = ( c ⊥ ) ∗ is a σ -ideal of the lattice L ( A ). Applying Proposition 8.16 it follows that L ( A ) is aconormal lattice. Theorem 8.19
For a coherent quantale A the following are equivalent:(1) A is a P F - quantale;(2) A is a semiprime mp - quantale. Proof. (1) ⇒ (2) By Proposition 8.18 and Corollary 8.10.(2) ⇒ (1) In order to prove that A is a P F -quantale let us assume that c ∈ K ( A ). We shall prove that c ⊥ is a pure element of A . Let d be a compactelement of A such that d ≤ c ⊥ , hence λ A ( c ) ∧ λ A ( d ) = λ A ( cd ) = λ A (0) = 0, i.e. λ A ( d ) ∈ Ann ( c ∗ ) = Ann ( λ A ( c )). By Corollary 8.10 L ( A ) is a conormal lattice,hence Ann ( λ A ( c )) is σ - ideal of L ( A ) (cf. Proposition 8.16). It follows that Ann ( c ∗ ) ∨ Ann ( d ∗ ) = Ann ( λ A ( c )) ∨ Ann ( λ A ( d )) = L ( A ).According to Lemma 3.4, Proposition 4.5 and Corollary 3.10, the followingequalities hold: ρ ( ρ ( c ⊥ ) ∨ ρ ( d ⊥ )) = ρ ((( c ⊥ ) ∗ ) ∗ ∨ (( c ⊥ ) ∗ ) ∗ ) = ρ (( Ann ( c ∗ )) ∗ ∨ ( Ann ( d ∗ )) ∗ ) =( Ann ( λ A ( c )) ∨ Ann ( λ A ( d ))) ∗ = ( L ( A )) ∗ = 1.By Lemma 2.2,(3) and (6) one gets c ⊥ ∨ d ⊥ = 1, hence c ⊥ is pure. Theorem 8.20
For a coherent quantale A consider the following conditions:(1) Any minimal m - prime element of A is pure;(2) A is an mp - quantale.Then (1) implies (2) . If the quantale A is semiprime then the converse impli-cation holds. Proof.
First we shall prove that (1) implies (2). According to Corollary 8.10it suffices to check that the reticulation L ( A ) is a conormal lattice. Let P be a minimal prime ideal of L ( A ), hence P = p ∗ for some p ∈ M in ( A ). Bytaking into account the hypothesis, it results that p is a pure element of A . Inaccordance to Lemma 4.7, P = p ∗ is a σ - ideal of L ( A ), so any minimal primeideal of L ( A ) is a σ - ideal. Applying Proposition 8.16 it follows that the lattice L ( A ) is conormal. 27ssume now that A is a semiprime mp - quantale and p ∈ M in ( A ), so p ∗ isa minimal prime ideal of L ( A ). By Corollary 8.10, L ( A ) is a conormal lattice,thus any minimal prime ideal of L ( A ) is a σ - ideal. Therefore p ∗ is a σ -ideal of L ( A ). Since A is semiprime, by applying Proposition 4.7 it follows that p = ( p ∗ ) ∗ is a pure element of A . Corollary 8.21
Let A be a semiprime quantale. Then A is a P F - quantale ifand only if any minimal m - prime element of A is pure. Proof.
We apply Theorems 8.19 and 8.20.
Theorem 8.22
For a coherent quantale A the following are equivalent:(1) A is a P F - quantale;(2) A is a semiprime mp - quantale;(3) If c, d ∈ K ( A ) then cd = 0 implies c ⊥ ∨ d ⊥ = 1 ;(4) If c, d ∈ K ( A ) then ( cd ) ⊥ = c ⊥ ∨ d ⊥ ;(5) For each c ∈ K ( A ) , c ⊥ is a pure element. Proof.
The equivalence of (1) and (2) follows from Theorem 8.19 and that theconditions (3) and (5) are equivalent is obvious.(2) ⇒ (3) Assume by absurdum that there exist c, d ∈ K ( A ) such that cd = 0and c ⊥ ∨ d ⊥ <
1, hence there exists a minimal m - prime element p such that p ≤ m . Since cd = 0 and p ∈ Spec ( A ) we have c ≤ p or d ≤ p . Assume that c ≤ p , so p ∨ c ⊥ = 1 (by Theorem 8.20, the minimal m - prime element p ispure). This contradicts p ∨ c ⊥ ≤ m , so the implication is proven.(3) ⇒ (2) First we prove that A is semiprime. Let c ∈ K ( A ) such that c n = 0for some integer n ≥
1. Assuming n >