EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 1, 2023, 314-318 ISSN 1307-5543 – ejpam.com Published by New York Business Global Graphs and the prime spectrum of unitary commutative rings Badr AlHarbi Umm Al-Qura University, Al Jumum University College, Department of Mathematics Abstract. In This paper, we study the relationships between graphs and the prime spectrum of unitary commutative rings. It is shown that a graph G equipped with the G-right topology satisfies some spectral properties. In particular we give a necessarily and sufficient condition to obtain a spectral graph. 2020 Mathematics Subject Classifications: 54F65, 54H20 Key Words and Phrases: Graph, spectral, prime spectrum, ring, Alexandroff space 1. Introduction In [4], Hochster proved that an ordered set (Y,≤) is order-isomorphic to the prime spectrum of a commutative ring with unit equipped with the inclusion if and only if the set Y is equipped with a topology compatible with the order and satisfying the following properties: i) X is a quasi-compact space. ii) X is a T0-space. iii) Each irreducible closed subset has a generic point. iv) X has a basis of quasi-compact open subsets. v) The intersection of two quasi-compact open subsets is quasi-compact. The above five properties are called spectral properties. Note that a topology compat- ible with the order is always T0 [2]. A topology defined on the set Y satisfying the properties i), iii), iv) and v) is called a quasi-spectral topology. DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4654 Email address: bhharbi@uqu.edu.sa (B. AlHarbi) https://www.ejpam.com 314 © 2023 EJPAM All rights reserved. B. AlHarbi / Eur. J. Pure Appl. Math, 16 (1) (2023), 314-318 315 • The space X is said to be quasi-compact if it satisfies the property of Borel-Lebesgue but it is not necessarily a Hausdorff space. • A topological spaceX is a T0-space (or Kolmogorov space) if for every pair of distinct points x and y, there exists a neighborhood containing one of them but not the other; which is equivalent to the following implication ({x} = {y} ⇒ x = y). • A closed subset C is irreducible if it is not the union of two proper closed subsets or if the intersection of two nonempty open subsets is nonempty. An element x of C is called a generic point if the closure of the singleton {x} is equal to C: {x} = C. We have the following properties: 1. The quasi-compactness is invariant under continuous map. 2. Each closed subset of a quasi-compact space is quasi-compact. 3. The union of finitely many quasi-compact subsets is quasi-compact. The intersection of tow quasi-compact open subsets is not necessarily quasi-compact. [1, Example 2.1] confirm this result. By [5], a spectral set satisfies the following conditions: (K1) Each totally ordered family of elements in (Y,≤) has a supremum and an infi- mum. (K2) For every elements a < b in Y , there exist two consecutive elements a1 < b1 with a ≤ a1 < b1 ≤ b. Lewis and Ohm showed in [6] that these two conditions are not sufficient to charac- terize ordered spectral sets. They even added a third independent of (K1) and (K2) (still necessary not sufficient): (H) Let F be a subset of L = {] ←, x] : x ∈ X} or R = {[x,→ [: x ∈ X}. If⋂ f∈F f = ∅, then F contains a finitely many elements with empty intersection. Where ]←, x] = {y ∈ X|y ≤ x} and [x,←, x[= {y ∈ X|x ≤ y}. Note that the problem of characterization of spectral set is still open. In this paper we define and characterise spectral graph. 2. Quasi-homeomorphism and spectral properties According to [3], a continuous mapping f : X → Y between two topological spaces is a quasi-homeomorphism if the map which associates to each open subset V ⊂ Y the open subset U = f−1(V ) ⊂ X is a bijective mapping. Equivalently, the map which assigns to each closed subset G ⊂ Y the closed subset F = f−1(G) ⊂ X is also a bijective mapping. We have the following properties: Let f : X → Y be a quasi-homeomorphism. 1. The composition of two quasi-homeomorphisms is a quasi-homeomorphism. B. AlHarbi / Eur. J. Pure Appl. Math, 16 (1) (2023), 314-318 316 2. f is open, closed. 3. For every locally closed subset A ⊂ X, we have A = f−1(f(A)). We say that every locally closed subset of X is f -saturated. 4. For every x, y ∈ X, we have the following implication: f(x) = f(y) ⇒ {x} = {y} 5. If moreover X is a T0-space, then f is an embedding (f : X → f(X) is a homeomor- phism). Theorem 2.1. If f : (X,T ) → (Y, T ′) is a onto quasi-homeomorphism, then T is quasi- spectral if and only if T ′ is quasi-spectral. Proof. We start by showing the following: Let f : X → Y be a quasi-homeomorphism and S be a subset of Y . 1. If S is an open set, then S is quasi-compact in Y if and only if, f−1(S) is quasi- compact in X. 2. If S is a closed set, then S is irreducible in Y if and only if, f−1(S) is irreducible in X. 1. Suppose that S is a quasi-compact open subset in Y . Let (Ui, i ∈ I) be an open covering of f−1(S). The fact that f is a quasi-homeomorphism implies that, for each i ∈ I, there exist an open subset Vi of Y such that Ui = f−1(Vi). Therefore f−1(S) = f−1( ⋃ i∈I Vi) and so S = ⋃ i∈I Vi. It follows from the fact that S is quasi- compact in Y , that there exists a finite subset J of I such that S = ⋃ i∈J Vi, which gives f−1(S) = ⋃ i∈J Ui and so f−1(S) is quasi-compact in X. Conversely, Suppose that f−1(S) is a quasi-compact open subset inX. Let (Vi, i ∈ I) be an open covering of S. Then f−1(S) = ⋃ i∈I f −1(Vi) and so there exists a finite subset J of I such that f−1(S) = ⋃ i∈J f−1(Vi) = f−1( ⋃ i∈J Vi) Using the fact that f is a quasi-homeomorphism we obtain S = ⋃ i∈J Vi and so S is quasi-compact in Y . 2. Suppose that S is an irreducible closed subset of Y . Let F and K be two closed subset of X such that f−1(S) = F ∪ K. Since f is a quasi-homeomorphism there exists two closed subsets F ′ and K ′ of Y such that f−1(F ′) = F and f−1(K ′) = K. Hence f−1(S) = f−1(F ′ ∪ K ′), which gives S = F ′ ∪ K ′. From the fact that S is an irreducible closed subset of Y , it follows that S = F ′ or F = K ′, this yields f−1(S) = F or f−1(S) = K. Therefore f−1(S) is an irreducible closed subset of X. B. AlHarbi / Eur. J. Pure Appl. Math, 16 (1) (2023), 314-318 317 Conversely, let F ′ and K ′ be two closed subset of Y such that S = F ′ ∪K ′. Then f−1(S) = f−1(F ′) ∪ f−1(K ′) and from the fact that f−1(S) is an irreducible closed subset of X it follows that f−1(S) = f−1(F ′) or f−1(S) = f−1(K ′). Since f is a quasi-homeomorphism, S = F ′ or S = K ′ which implies that S is an irreducible closed subset of Y . IfX is quasi-compact, then since f is onto and continuous, f(X) = Y is quasi-compact. By the above item (1), if Y is quasi-compact, then f−1(Y ) = X is quasi-compact. By the above item (1), (X,T ) has a base of quasi-compact open subsets if and only if (Y, T ′) has a base of quasi-compact open subsets. By the above item (1) and the fact that f is onto and continuous, the family of quasi- compact open subsets of (X,T ) is stable by finite intersection if and only if the family of quasi-compact open subsets of (Y, T ′) is stable by finite intersection. By the above item (2) and the fact that f is onto and continuous, every irreducible closed subset of (X,T ) has a generic point if and only if every irreducible closed subset of (Y, T ′) has a generic point. This ends the proof of the theorem. 3. Spectral graph Let G = (V,E) be a graph (finite or infinite) and let u, v ∈ V . A path from u to v in G is a sequence of edges e1, . . . , en of E for which there exists a sequence x0 = u, x1, . . . , xn−1, xn = v of vertices such that ei has, for i = 1, ..., n, the endpoints xi−1 and xi. We denote by R(u) = {u} ∪ {v : if there exists a path from u to v} L(u) = {u} ∪ {v : if there exists a path from v to u}. The family {R(u) : u ∈ G} (respectively {L(u) : u ∈ G}) forms a base of a topology on G called the G-right τ(GR) (respectively G-left τ(GL)) topology. Two vertices a and b in a graph G are called adjacent in G if a and b are endpoints of an edge e of G. The graphs G1 = (V1, E1) and G2 = (V2, E2) are isomorphic if there exists a one-to-one and onto function f from V1 to V2 with the property that a and b are adjacent in G1 if and only if f(a) and f(b) are adjacent in G2, for all a and b in V1. Such a function f is called an isomorphism. Definition 3.1. [1] The graphs G1 = (V1, E1) and G2 = (V2, E2) are homeomorphic if (G1, τ(G R 1 )) and (G2, τ(G R 2 )) are homeomorphic. Note that, according to [1], if G1 = (V1, E1) and G2 = (V2, E2) are isomorphic, then they are homeomorphic. If moreover every vertex of G1 and G2 has a loop, then isomorphic and homeomorphic properties are equivalent. REFERENCES 318 Let G = (V,E) be a graph equipped with a topology T . We say that T is compatible with the graph structure G, if for all u ∈ V , {u} = R(u). We remark that the G-left τ(GL) topology is the finer topology compatible with the graph structure G. We define on G the following relation u ⪯ v if u ∈ R(v). It is easy to see that ⪯ is reflexive and transitive. Then ⪯ is a preorder. We define an equivalence relation on G by uRv if and only if R(u) = R(v). The quotient set (the set of equivalence classes) is denoted by G/R. (G/R,⪯) is a pre-ordered set. Let G/R̃ be the universal T0-space associated to the space G/R as in Bourbaki [2, Exercise 27 p: 1-104]. Definition 3.2. G is a spectral graph if there exists a quasi-spectral topology T compatible with the graph structure G. Theorem 3.3. G is a spectral graph if and only if (G/R̃,⪯) is order-isomorphic to the prime spectrum of a unitary commutative ring equipped with the inclusion. Proof. Let projection q : (G,T ) → G/R be the canonical. Let T be the quotient topology on G/R. Let φ : T → T be the defined by φ(U) = q−1(U), for all U ∈ T . First, we show that φ is onto. It suffices to show that q−1(q(U)) = U . It is easy to see that U ⊂ q−1(q(U)). Let x ∈ q−1(q(U)). Then q(x) ∈ q(U) which implies that there exits y ∈ U such that q(x) = q(y). Therefore R(x) = R(y). Since T is compatible with G, R(x) ⊂ U which implies that x ∈ U . Thus q−1(q(U)) = U . Second, since q is onto, we get q(q−1(V )) = V for all V ∈ T . Then φ is injective. Consequently, φ is bijective and so q is an onto quasi-homeomorphism. By a same method as above we get ψ : G/R → G/R̃ which associates to each R(u) its class R̃(u) = {v ∈ G : {R(u)} = {R(v)}} is a quasi-homeomorphism. Therefore ψ ◦ q : (G,T )→ G/R̃ is a quasi-homeomorphism. Hence, By Theorem 2.1 we get (G,T ) to be quasi-spectral if and only if G/R̃ is quasi- spectral. Since G/R̃ is a T0-space we obtain Theorem 3.3. Note that a finite graph is spectral. References [1] B. Alharbi. Graphs and alexandroff spaces. Submitted. [2] N. Bourbaki. General topology chapter 1 to 4. Masson, New York, 1990. [3] A. Grothendieck and J. Dieudonné. Éléments de Géométrie Algébrique. Springer- Verlag, New York, 1971. [4] M. Hochster. Prime ideal structure in commutative rings. Trans. Amer. Math. Soc., 142:43–60, 1969. [5] L. Kaplansky. Graphs and Alexandroff spaces (Revised edition). The Universty of Chicago, Press, 1974. [6] W.J.Lewis and J.Ohm. The ordring of spec. R.Can.J.Math Vol, 28:820–835, 1973.