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Stone Type Representations and Dualities by Power Set Ring

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Stone Type Representations and Dualities by Power Set Ring STONE TYPE REPRESENTATIONS AND DUALITIES BY POWER SET RING ABOLFAZL TARIZADEH AND ZAHRA TAHERI Abstract. In this paper, it is shown that the Boolean ring of a commutative ring is isomorphic to the ring of clopens of its prime spectrum. In particular, Stone’s Representation Theorem is gen- eralized. The prime spectrum of the Boolean ring of a given ring R is identified with the Pierce spectrum of R. The discreteness of prime spectra is characterized. It is also proved that the space of connected components of a compact space X is isomorphic to the prime spectrum of the ring of clopens of X. As another ma- jor result, it is shown that a morphism of rings between complete Boolean rings preserves suprema if and only if the induced map between the corresponding prime spectra is an open map. Contents 1. Introduction 1 2. Preliminaries 3 3. Generalization of Stone Representation Theorem 4 4. Booleanization versus Pierce spectrum 6 5. Power set ring and Stone duality 8 6. Complete Boolean rings and Stone type duality 11 7. Power set ring and fixed-point theorems 15 References 19 arXiv:1905.10612v6 [math.AC] 14 Feb 2021 1. Introduction In 1936, Marshall Stone published a long paper [14] that whose main result was that every Boolean ring is isomorphic to a certain subring of a power set ring. This Stone Representation Theorem has been the starting point for a whole new field of study, nowadays called Stone duality. It is worth mentioning that “Boolean ring” and “Boolean 2010 Mathematics Subject Classification. 14A05, 06E15, 13C05, 13A15, 13M05, 06E05, 06E20, 54H25. Key words and phrases. Power set ring; Boolean ring; Clopen; Complete Boolean ring; Galois connection; Stone type duality. 1 2 A.TARIZADEHANDZ.TAHERI algebra” are the equivalent structures, i.e., every Boolean ring can be made canonically into a Boolean algebra (and vice versa). This is a well known fact, for its proof and further studies on these topics we refer the interested reader to the References, especially to [3], [7], [8] and [11]. In this paper we prove that if R is a commutative ring, then we have the canonical isomorphism of rings: B(R) ≃ Clop Spec(R) here B(R) denotes the ring of idempotents of R; we call B(R) the Boolean ring of R. This is Theorem 3.1. Note that the Stone space of a Boolean algebra (see e.g. [3, Proposition 4.1.11]) and the prime spectrum of the corresponding Boolean ring are canonically homeo- morphic. In fact, this homeomorphism takes each ultrafilter F in a given Boolean algebra R into R \ F which is a maximal ideal of the corresponding Boolean ring. By taking into account this identification, then the original Stone’s Representation Theorem is a especial case of the above isomorphism, see Corollary 3.2. Indeed, considering Clop(X) as a ring (where X is a topological space) is the simple idea leading to some results of this paper. In Theorem 5.3, we prove the topological version of Theorem 3.1 which states that if X is a compact space then we have the canonical isomorphism of topological spaces: π (X) ≃ Spec Clop(X) 0 where π0(X) denotes the space of connected components of X which carries the quotient topology. If moreover X is totally disconnected, then we easily get the following well known isomorphism which can be considered it as the topological version of the Stone’s Representation Theorem: X ≃ Spec Clop(X) . In Sections 3 and 4 we prove several structural results on the Boolean ring of a given ring and its prime spectrum. Especially, the structure of prime ideals of the Boolean ring of a given ring R is identified in terms of the connected components of Spec(R). In fact, in Theorem 4.1, we obtain the following canonical isomorphism of topological spaces: Spec B(R) ≃ π Spec(R) . 0 Then as an application, for a given ring R, it is shown that X = Spec(R) is discrete if and only if B(R) ≃ P(X), see Theorem 4.2. Sections 5 and 6 investigate the structure of Boolean rings, especially complete Boolean rings. Theorem 5.7 gives a canonical bijection from STONE TYPE REPRESENTATIONS AND DUALITIES 3 the set of ring maps between two Boolean rings onto the set of contin- uous maps between their corresponding prime spectra. In Theorem 6.3, we prove a technical result which states that a mor- phism of rings between complete Boolean rings preserves suprema if and only if the induced map between the corresponding prime spectra is an open map. This observation leads us to various non trivial results. Especially among them, we obtain a Stone type duality which states that the category of complete Boolean rings is the dual of the category of compact extremally disconnected spaces, see Theorem 6.5. In §7, we have made further progresses in the understanding the struc- ture of power set ring. Also some contributions to the field of fixed- point theorems are provided. 2. Preliminaries We collect in this section some basic background for the reader’s convenience. In this paper, all rings are commutative. If f is an element of a ring R, then D(f)= {p ∈ Spec(R): f∈ / p} and V(f) = Spec(R) \ D(f). If X is a set then its power set P(X) together with the symmetric dif- ference A + B =(A ∪ B) \ (A ∩ B) as the addition and the intersection A.B = A ∩ B as the multiplication forms a commutative ring whose zero and unit are respectively the empty set and the whole set X. The ring P(X) is called the power set ring of X. If ϕ : X → Y is a function then the map P(ϕ): P(Y ) → P(X) defined by A ϕ−1(A) is a mor- phism of rings. In fact, the assignments X P(X) and ϕ P(ϕ) form a faithful contravariant functor from the category of sets to the category of commutative rings. We call it the power set functor. Remember that a ring is called a Boolean ring if each element is an idempotent. Power set ring P(X) is a typical example of Boolean rings. It is easy to see that in a Boolean ring, every prime ideal is a maximal ideal. By a compact space we mean a quasi-compact and Hausdorff topologi- cal space. The number of elements (cardinal) of a set X is denoted by |X|. If ϕ : R → R′ is a morphism of rings then the induced map Spec(R′) → Spec(R) defined as p ϕ−1(p) is denoted by Spec(ϕ). If A and B are objects of a category C , then as usual by MorC (A, B) we mean the set of morphisms of C from A to B. By a join-complete poset we mean a poset A such that the supremum of every subset (xk) ⊆ A exists in A, this element is denoted by xk. Wk For example the posets P(X) and Spec(Z) ordered by the inclusion, 4 A.TARIZADEHANDZ.TAHERI are join-complete and not join-complete, respectively. Dually, by a meet-complete poset is meant a poset A such that the infimum of every subset (xk) ⊆ A exists in A, this element is denoted by xk. Recall that a poset which is both join-complete and meet- Vk complete is called a complete lattice. It is easy to see that an order preserving map f : A → B between the join-complete posets preserves suprema if and only if there exists an order preserving map g : B → A such that the pair (f,g) is a Galois connection, i.e., f(a) 6 b ⇔ a 6 g(b) for all a ∈ A and b ∈ B. In this case, such function g is unique. In fact, if f preserves suprema, then g : B → A is defined as g(b)= a. f(Wa)6b Dually, an order preserving map f : A → B between the meet-complete posets preserves infima if and only if there exists an order preserving map g : B → A such that the pair (g, f) is a Galois connection. If f preserves infima, then g(b)= a. f(Va)>b 3. Generalization of Stone Representation Theorem In this section, Stone’s Representation Theorem is generalized from Boolean rings to arbitrary commutative rings. If X is a topological space then by Clop(X) we mean the set of clopen (both open and closed) subsets of X. It has the canonical ring struc- ture. In fact, Clop(X) is a subring of P(X). If ϕ : X → Y is a con- tinuous map of topological spaces then the map Clop(ϕ) : Clop(Y ) → Clop(X) given by A ϕ−1(A) is a morphism of rings. In fact, the as- signments X Clop(X) and ϕ Clop(ϕ) form a contravariant func- tor from the category of topological spaces to the category of Boolean rings. We call it the Clopen functor. Let R be a ring and B(R)= {e ∈ R : e = e2}. Then it is easy to see that the set B(R) by the new operation ⊕ defined by e⊕e′ := e+e′ −2ee′ as the addition admits a ring structure whose multiplication is the multi- plication of R. Note that B(R) is not necessarily a subring of R. In fact, B(R) is a subring of a non-zero ring R if and only if Char(R) = 2. We call B(R) the Boolean ring (or, Booleanization)of R.
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