The frame of d-ideals of RL

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Characterisation of z-ideals of RL

Following [58] we define for any a ∈ A the set M(a) = {M ∈ Max A | a ∈ M}. An ideal I of A is called a z-ideal in case for any a, b ∈ A, a ∈ I and M(a) = M(b) imply b ∈ I. This is equivalent to saying a ∈ I and M(a) ⊇ M(b) imply b ∈ I. It is apposite to remark that although a number of authors seem to attribute this definition to Mason (emanating from his paper [58]), it already appears in Kohl’s paper [49]. As mentioned above, Dube [33] showed that an ideal Q of RL is a z-ideal if and only if for any α, β ∈ RL, α ∈ Q and coz α = coz β imply β ∈ Q. Here we give an alternative proof which brings to the fore a number of other noteworthy observations. We start with the following lemma.

Zid(RL) is coherently normal.

We show next that Zid(RL) has a property which is a stronger version of normality. In [10], Banaschewski calls a frame L coherently normal if it is coherent and, for each compact c ∈ L, the frame ↓c is normal. We show below that Zid(RL) is coherently normal. We will use [19, Lemma 1] which (paraphrased) states: For any elements a and b of a σ-frame L, there exist u and v in L such that u ∧ v = 0 and a ∨ u = b ∨ v = a ∨ b. In the proof of this result it is clear that u ≤ b and v ≤ a. An algebraic frame L is said to have the finite intersection property (abbreviated FIP) if the meet of any two compact elements in L is compact. Mart´ınez [52] says an algebraic frame L has disjointification – a property equivalent to coherent normality for algebraic frames with FIP – if for each pair of compact elements a, b ∈ L, there exist disjoint compact elements c ≤ a and d ≤ b in L with a ∨ b = a ∨ d = b ∨ c, and remarks that if L has FIP, then it is coherently normal if and only if it has disjointification. Let us observe an easy lemma for use in the upcoming result and later.

A note on flatness

We remind the reader that a frame homomorphism h: L → M is flat if h is onto and h∗ : M → L is a lattice homomorphism [13]. Weakening this, we say h is coz-flat if h∗(0) = 0 and h∗(a ∨ b) = h∗(a) ∨ h∗(b) for all a, b ∈ Coz L. Observe that coz-flatness is a genuine weakening of flatness. Indeed, for any non-normal completely regular frame L, the join map βL → L is coz-flat, but not flat. We aim to show that for a homomorphism h whose right adjoint sends cozero elements to cozero elements, Zid(h) is flat precisely when h is coz-flat. We need a lemma. Lemma 2.5.1. Let h: L → M be a morphism in CRegFrm. For all S, T ∈ Zid(RL) and Q, R ∈ Zid(RM) we have: (1) S ∨ T = W {Mcoz γ | γ ∈ S + T} = S {Mcoz γ | γ ∈ S + T} (2) Zid(h)∗(Q ∨ R) = W {Mh∗(coz τ) | τ ∈ Q + R}. Proof. (1) Observe that the join is directed, and hence equals the union. The rest is easy to check.

Contracting z-ideals

In this section we investigate if z-ideals of RL contract to z-ideals of R∗L, and if z-ideals of the smaller ring extend to z-ideals of the bigger ring. Recall that if φ: A → B is a ring homomorphism and I is an ideal of B, then φ −1 [I] is an ideal of A called the contraction of I, and frequently denoted by I c . On the other hand, if J is an ideal of A, the (possibly improper) ideal of B generated by φ[J] is called an extension of J and denoted by J e . In the event that A is a subring of B and φ the inclusion map, then I c = I ∩ A. In what follows we shall make use of the well-known fact that R∗L ∼= R(βL) [22], and R(βL) ∼= C(X) for some topological space X. For later use, let us recall how a ring isomorphism can be constructed. (1) If h: L → M is a dense frame homomorphism, then the ring homomorphism Rh: RL → RM is one-one ([14, Lemma 2]). (2) The frame homomorphism jL : βL → L is, in the terminology of [6], a C ∗ -quotient map ([6, Corollary 8.2.7]), meaning that for every α ∈ R∗L there is a (necessarily unique) element of R(βL), which, as in the classical case [39], we shall denote by α β , such that the triangle

Coherence of the frame of d-ideals of an f-ring

We recall that an ideal of a ring A is singular if it consists entirely of zero-divisors. For any a ∈ A, let Pa denote the intersection of all minimal prime ideals of A containing a. It is shown in [59] that Pa = Ann2 (a). An ideal I of A is called a d-ideal if Ann2 (a) ⊆ I, for every a ∈ I. Examples of d-ideals abound (see, for instance, [4]). It is clear that the union of a directed family of d-ideals is a d-ideal. As stated in the Introduction, we shall at times write the annihilator of a set S as S ⊥, and that of an element a as a ⊥. Lemma 3.1.1. Let A be a reduced f-ring and I be a singular ideal of A. Then the set J = [ {a ⊥⊥ | a ∈ I} is the smallest d-ideal of A containing I. Proof. Let us show first that the family {a ⊥⊥ | a ∈ I} is directed. Let a, b ∈ I. We claim that a ⊥⊥ ∪ b ⊥⊥ ⊆ (a 2 + b 2 ) ⊥⊥. To verify this it suffices to show that (a 2 + b 2 ) ⊥ ⊆ a ⊥ ∩ b ⊥. Let r ∈ (a 2 + b 2 ) ⊥. Then r(a 2 + b 2 ) = 0, which implies (ra) 2 + (rb) 2 = 0. Since squares are positive in f-rings, this implies (ra) 2 = (rb) 2 = 0, and hence ra = rb = 0 since A is reduced. Therefore r ∈ a ⊥ ∩ b ⊥. Thus, J is d-ideal which clearly contains I. To show that it is the smallest such, consider any d-ideal K of A which contains I. Let u ∈ J. Then u ∈ a ⊥⊥ for some a ∈ I.

Contents :

• Declaration
• Acknowledgements
• 1 Introduction and preliminaries
  • 1.1 A brief history on z-ideals and d-ideals
  • 1.2 Synopsis of the thesis
  • 1.3 Frames and their homomorphisms
  • 1.4 Rings and f-rings
  • 1.5 The rings RL and R∗L
• 2 The frame of z-ideals of RL
  • 2.1 Characterisation of z-ideals of RL
  • 2.2 The frame Zid(RL)
  • 2.3 Zid(RL) is coherently normal
  • 2.4 Some commutative squares associated with z-ideals
  • 2.5 A note on flatness
  • 2.6 Contracting z-ideals
• 3 The frame of d-ideals of an f-ring
  • 3.1 Coherence of the frame of d-ideals of an f-ring
  • 3.2 Extending and contracting d-ideals
• 4 The frame of d-ideals of RL
  • 4.1 Characterisation of d-ideals of RL
  • 4.2 The frame Did(RL)
  • 4.3 Projectability properties
• 5 Two functors induced by z- and d-ideals
  • 5.1 The functors Z and D
  • 5.2 Replacing morphisms with their right adjoints
  • 5.3 Some commutative squares associated with d-ideals
  • 5.4 Preservation and reflection of certain properties
  • 5.5 Preservation and reflection of openness by the functor Z
• 6 Covering maximal ideals
  • 6.1 Variants of roundness
  • 6.2 When RL is a UMP-ring
• 7 A miscellany of results
  • 7.1 Existence of nth roots in RL
  • 7.2 Some other properties of z-ideals
  • 7.3 On the Stone-Cech compactification of frames

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