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@ Appl. Gen. Topol. 19, no. 2 (2018), 217-222doi:10.4995/agt.2018.7952
c© AGT, UPV, 2018

Topological characterization of Gelfand and

zero dimensional semiring

Jorge Vielma and Luz Marchan
∗

Escuela Superior Politécnica del Litoral. ESPOL, FCNM, Campus Gustavo Galindo Km. 30.5 V́ıa

Perimetral, P.O.Box 09-01-5863. Guayaquil, Ecuador (jevielma@espol.edu.ec, lmarchan@espol.edu.ec)

Communicated by J. Galindo

Abstract

Let R be a conmutative semiring with 0 and 1, and let Spec(R) be
the set of all proper prime ideals of R. Spec(R) can be endowed with
two topologies, the Zariski topology and the D-topology. Let MaxR

denote the set of all maximal prime ideals of R. We prove that the two

topologies coincide on Spec(R) and on MaxR if and only if R is zero
dimensional and Gelfand semiring, respectively.

2010 MSC: 54A10; 54F65; 13C05; 16Y60.

Keywords: Zariski topology; D-topology; conmutative semiring; Gelfand

semiring; zero dimensional semiring.

1. Basic facts

Recall that a semiring (conmutative with non zero identity) is an algebra
(R, +, ·, 0, 1), where R is a set with 0, 1 ∈ S, and + and · are binary operations
on R called sum and multiplication, respectively, which satisfy the following:

(1) (R, +, 0) and (R, ·, 1) are conmutative monoid with 1 6= 0.
(2) a · (b + c) = a · b + a · c for every a, b, c ∈ R.
(3) a · 0 = 0 for every a ∈ R.

∗The authors are supported by the research project ESPOL FCNM-09-2017.

Received 22 August 2017 – Accepted 09 February 2018

http://dx.doi.org/10.4995/agt.2018.7952


J. Vielma and L.Marchan

A subset I of R will be called an ideal of R if a, b ∈ I and r ∈ R implies
a + b ∈ I and ra ∈ I. A prime ideal of R is a proper ideal P of R in which
x ∈ P or y ∈ P whenever xy ∈ P . The nilradical of R, denoted by N(R), is
the intersection of all the prime ideals de R. Max(R) and Min(R) denote the
set of all maximal and minimal prime ideals of R, respectively. R is said to be
Gelfand if every prime ideal is contained in at most one maximal ideal. R is
said to be zero dimensional if every prime ideal of R is maximal.

For x ∈ R, let (0 : x) = {y ∈ R : xy = 0}. An ideal I of R is called a ó-ideal
if for every x ∈ R, I(0 : x) = R, that is to say there exist x1 ∈ I and y ∈ (0 : x)
such that 1 = x1y. A element x ∈ R is called a complemented element in R if
there is y ∈ R such that xy = 0 and x + y = 1, y is called the complement of
x.

For any semiring R, Spec(R) denotes the set of all proper prime ideals of
R. This set can be given the Zariski topology τz as follows: For every proper
set I of R, let (I)0 = {P ∈ Spec(R) : I ⊆ P} and let D(I) = Spec(R) \ (I)0 =
{P ∈ Spec(R) : I * P}. If I is the ideal generated by a ∈ S, we write I = (a).
Note that (a)0 = {P ∈ Spec(R) : a ∈ P} and D(a) = {P ∈ Spec(R) : a /∈ P}.
The sets D(a) ⊆ Spec(R) with a ∈ R, constitute a basis for τz, and the sets
(I)0 with I ideal of R are the closed sets for τz.

Let (X, τ) a topological space, τ∗ denote the family of τ-closed subset of
X. τ is said to be Alexandroff if it is closed under arbitrary intersections. By
identifying a set with its characteristic function, we can view τ as a subset of
2X with the product topology, then its closure τ is also a topology, even more,

τ =

{

A ⊆ X : A =
⋂

θ∈L

θ, L ⊆ τ

}

and it is the smallest Alexandroff topology containing τz (see [5]).
Note that A ∈ τ∗ if and only if Ac is τ-open, let say Ac =

⋂

θ∈L
θ for some

L ⊆ τ, then

x ∈ A ⇒ x /∈ Ac ⇒ x /∈ θ, for some θ ∈ L

⇒ x ∈ θc, for some θ ∈ L

⇒ {x} ⊆ θc = θc

⇒ {x} ⊆

(

⋂

θ∈τ

θ

)c

= A

and A is just the union of the τ-closure of each of its points.

A subset A of a topological space (X, τ) is τ-saturated if {a} ⊆ A for all
a ∈ A, that is to say, if A ∈ τ∗. In particular, A ⊆ Spec(R) is τz-saturated if
and only if for each P ∈ A, (P)0 ⊆ A. The set of all τz-open and τz-saturated
subsets of Spec(R) defines a topology on Spec(R) called the D-topology, this is
to say that the D-topology is just τZ ∩ τZ

∗.
Remember that a topology τ on X is said T0 if for each pair of distinct

elements x and y in X, exist a open set containing either x or y, and τ is T1 if

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 218



Topological characterization of Gelfand and zero dimensional semiring

for each pair of distinct elements x and y in X, exist a open set containing x
and not y and an open set containing y and not x.

The following results, given in [2] and [5], characterize the topologies T0 and
T1 of the following manner:

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 219



J. Vielma and L.Marchan

Theorem 1.1. Let τ be a topology on X then,

(i) τ is T0 if and only if τ is T0.
(ii) τ is T0 if and only if τ ∨ τ

∗ = ℘(X)
(iii) τ is T1 if and only if τ = ℘(X).

In [1], Al-Ezeh endowed Spec(L), where L is a distributive lattice with 0
and 1, with two topologies, the τz-topology and the D-topology, and he proved
that this two topologies coincide on Spec(L) and Max(L) iff L is a boolean
and normal lattice, respectively. In [3], Rafi and Rao introduced the concept
of D-topology on Spec(R), where R is a almost distributive lattice (ADL),
and characterized those ADLs for which topologies coincide on Spec(R) and
Min(R). In this paper, the concept of D-topology is introduced on Spec(R),
where R is a semiring, we do a similar study, as a consequence, we obtain a
result given in [1] for distributive lattices.

2. Main results

We begin by establishing some relationships between the τz-open sets and
D-open and between the τz-clopen and the D-clopen.

Remark 2.1. If I is a ó-ideal of a semiring R, then D(I) is D-open. In effect,
let P ∈ D(I), we will show that (P)0 ⊆ D(I). Let Q ∈ (P)0, since P ∈ D(I),
I * P , hence exists x ∈ I such that x /∈ P . Since I is a ó-ideal, there exist
x1 ∈ R and y ∈ (0 : x) such that x1 + y = 1, note that y ∈ P ⊆ Q (because
xy = 0 ∈ P and x /∈ P) so x1 /∈ Q (otherwise 1 = x1 + y ∈ Q) implying I * Q,
in consequence, Q ∈ D(I).

The reciprocal of the previous remarks it is not true, as shown in the follow-
ing example.

Example 2.2. Let A a non-empty subset of a set X, and let L = {∅, A, Ac, X},
(L, ∪, ∩) is a semiring where the sum and multiplication are the union and
intersection, respectively, and the identities of the sum and multiplication are
the empty set and the whole set X, even more (L, ∪, ∩) is a distributive lattice.
The ideals of L are {∅}, 〈A〉 = {∅, A}, 〈Ac〉 = {∅, Ac}, 〈X〉 = L. Spec(L) =
{〈A〉, 〈Ac〉} and D(〈A〉) = {〈Ac〉}, clearly D(〈A〉) is τ-saturated, but 〈A〉 it is
not a ó-ideal, since (∅ : Ac) = {∅, A} and 〈A〉 ∪ (∅ : Ac) = {∅, A} 6= L.

Proposition 2.3. Let R be a semiring with trivial nilradical and let I an
ideal of R. Then D(I) is D-clopen if and only if D(I) = D(x) for some
complemented element x in R.

Proof. Assume that D(I) is clopen. Then Spec(R) \ D(I) is also an open
set, so there exists an ideal J of R such that D(J) = Spec(R) \ D(I). Now
D(I) ∩ D(J) = D(IJ) = ∅, this implies IJ ⊆ P for all P ∈ Spec(R), this
is, IJ ⊆ N(R) = {0}. Also now, Spec(R) = D(I) ∪ D(J) = D(I + J), this
implies I + J = R, thus exist x ∈ I and y ∈ J such that x + y = 1. Since
IJ = {0}, we have xy = 0, then x is complemented. We see that I = (x), let
z ∈ I, z = z1 = z(x + y) = zx + zy = zx ∈ (x) since zy ∈ IJ = {0}. Thus I

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 220



Topological characterization of Gelfand and zero dimensional semiring

is a principal ideal generated by x, in consequence D(I) = D(x). Conversely,
assume that x is a complemented element in R, then there exists an element
y ∈ R such that xy = 0 and x+y = 1. Now D(x)∩D(y) = D(xy) = D(0) = ∅
and D(x)∪D(y) = D(x+y) = D(1) = Spec(R). Therefore D(x) is clopen. �

Now we characterize those semiring for which the Zariski topology and the
D-topology coincide on Spec(R).

Theorem 2.4. Let R be a semiring. Then the Zariski topology and the D-
topology coincide on Spec(R) if and only if R is zero dimensional.

Proof. Note that W is D-open if and only if W ∈ τz ∩ τz
∗, thus the Zariski

topology and the D-topology coincide on Spec(R) if and only if τz = τz ∩ τz
∗,

but

τz = τz ∩ τz
∗ ⇔ τz ⊆ τz

∗

⇔ τz ⊆ τz
∗

⇔ τz = τz
∗

⇔ τz ∨ τz
∗ = τz

⇔ τz = ℘(Spec(S)) (τz is T0 and by Theorem 1.1 part (i))

⇔ τz is T1 ( by Theorem 1.1 part (iii))

Now τz is T1 if and only if every P ∈ Spec(R) is closed impliying {P} = (P)0,
or equivalently, every prime ideal of R is maximal, this is, R is zero dimensional.

�

Theorem 2.5 ([4]). Let L be a distributive lattice. Then L is a boolean algebra
if and only if every prime ideal of L is a maximal ideal.

As a consequence of the Theorem 2.4 we obtain the following result given in
[1] for distributive lattices.

Corollary 2.6. Let L be a distributive lattice with 0 and 1. Then the Zariski
topology and the D-topology coincide on Spec(R) if and only if L is a boolean
lattice (a lattice every element of which has a complement).

Proof. immediately of Theorem 2.4 and Theorem 2.5, since every lattice is a
semiring. �

Theorem 2.7. R is a Gelfand semiring if and only if τz and τz ∩ τz
∗ agree on

Max(R).

Proof. Suppose R is a Gelfand semiring. We want to prove that τz and τz ∩τz
∗

agree on Max(R). Since τz ∩ τz
∗ ⊆ τz, it remains to show that τz ⊆ τz ∩ τz

∗

on Max(R). So we want to show that each D(x) in τz restricted to Max(R)
is an open set in τz ∩ τz

∗ restricted to Max(R). For each x ∈ R, let D∗
x
=

D(x) ∩ Max(R) and let

W = {P ∈ Spec(R) : x /∈ MP },

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 221



J. Vielma and L.Marchan

where MP is the unique maximal ideal of R containing P . Let us prove then
that D∗

x
is a τz ∩ τz

∗-open subset in Max(R). Now, for each P ∈ W we have
that (P)0 ⊆ W , then W ∈ τ

∗
z
. Since D∗

x
= W ∩ Max(R) then, D∗

x
is a

τz ∩ τz
∗-open set in Max(R). Therefore the conclusion follows.

Conversely, suppose τz and τz ∩τz
∗ agree on Max(R) and take P ∈ Spec(R)

which is contained in two different maximal ideals M and N. Take, without
lost of generality, x0 ∈ M such that x0 /∈ N. So N ∈ D(x0) ∩ Max(R) =
W ∩ Max(R) for some τz ∩ τz

∗ open set W . Now since P ⊆ N and N ∈ W it
follows that (P)0 ⊆ W . Therefore (P)0 ∩ Max(R) ⊆ W ∩ Max(R) impliying
that x0 /∈ M, which is a contradiction. �

Question 2.8. Under what conditions on R, the τz-topology and the D-topology
coincide on Min(R)?

References

[1] H. Al-Ezeh, Topological characterization of certain classes of lattices, Rend. Sem. Univ.
Padova 83 (1990), 13–18.

[2] M. L. Colasante, C. Uzcátegui and J. Vielma, Boolean algebras and low separation
axioms, Topology Proceedings 34 (2009), 1–15.

[3] N. Rafi and G. C. Rao, Topological characterization of certain classes of almost distribu-
tive lattice, J. Appl. Math. & Informatics 33, no. 3–4 (2015), 317–325.

[4] M. T. Sancho, Methods of conmutative algebra for topology, Universidad de Salamanca,
Departamento de matemáticas, (1987).

[5] C. Uzcátegui and J. Vielma, Alexandroff topologies viewed as closed sets in the Cantor
cube, Divulg. Mat. 13, no. 1 (2005), 45–53.

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 222