1 Introduction

In this paper we study the nilcompactification from a categorical point of view. This is a compactification method of prime spectra presented in [3]. Generally, compactification is a topological process (see for example [4, 6, 10, 12]); however, nilcompactification is a method obtained mainly through an algebraic process. The categorical standpoint of nilcompactification, with some of its possible variations, gives an interesting approach to this process and several examples of different categorical concepts, that can be studied in depth in [9]. This method of nilcompactification is functorial in a simple way. Also, it has interesting properties and the involved constructions provide us with different natural transformations. To make a contrast, the classical method of Alexandroff compactification does not have, in general, a functorial behavior. Nevertheless, to obtain it, it is neccessary to consider suitable subcategories, as studied in [2]. This perspective inspired the work presented in sections four to six, with three variations in the functorial consideration of nilcompactification.

Along this paper the word ring means commutative ring, not necessarily with identity. A homomorphism is a function between rings that respects addition and product. We suppose that prime ideals are proper ideals by definition. The prime spectrum of a ring S is denoted by \(\textrm{Spec}S\) and is the set of prime ideals of S endowed with the Zariski topology. In this topology the sets

$$\begin{aligned} D_{S}(a)=\{P\in \textrm{Spec}S:a\notin P\}, \end{aligned}$$

where \(a\in S,\) provide a basis. The closed sets are

$$\begin{aligned} V_{S}(I)=\{P\in \textrm{Spec}S:P\supseteq I\}, \end{aligned}$$

where I is an ideal of S. It is known that the basic open sets are compact and for each ideal I of S the function \(V_{S}(I)\rightarrow \) \( \textrm{Spec}\left( S/I\right) :P\mapsto P/I\) is a homeomorphism (see [5]). The Zariski topology is also called the hull-kernel topology because the closure of a subset \({\mathcal {B}}\) of \(\textrm{Spec}S\) is the set

$$\begin{aligned} \left\{ P\in \textrm{Spec}S:P\supseteq \bigcap \limits _{J\in {\mathcal {B}} }J\right\} \end{aligned}$$

and \(\bigcap \nolimits _{J\in {\mathcal {B}}}J\) is called the kernel of \({\mathcal {B}} .\)

The nilradical of S,  denoted by \(N\left( S\right) ,\) is precisely the kernel of \(\textrm{Spec}S.\)

It is known that S is semiprime or reduced if \(N\left( S\right) =\left\{ 0\right\} .\)

A ring whose spectrum is compact is a spectrally compact ring. In particular, every unitary ring R is spectrally compact because \(D_{R}(1)= \textrm{Spec}R.\)

We use the following notations:

\({{\mathcal {C}}}{{\mathcal {R}}}:\) Category of commutative rings and homomorphisms of rings.

\({{\mathcal {C}}}{{\mathcal {R}}}_{1}:\) Category of unitary commutative rings and homomorphisms of unitary rings.

\({{\mathcal {C}}}{{\mathcal {R}}}^{s}:\) Category of commutative rings and surjective homomorphisms.

\(\textrm{Top}:\) Category of topological spaces and continuous functions.

\({\mathbb {S}}:\) Category of spectral spaces and strongly continuous functions. A spectral space is a topological space that is homeomorphic to the prime spectrum of a unitary commutative ring. It is known that a topological space is spectral if and only if it is sober, compact and coherent (see [7]). A function is strongly continuous if it sends compact open sets in compact open sets by reciprocal image.

2 Nilcompactification

The material of this section is taken from [3].

In this section S is a fixed ring and R is an i-extension of S,  that is, a ring containing S as ideal.

Given an ideal I of S it is clear that the set \(\psi (I)=\{x\in R:xS\subseteq I\}\) is an ideal of R. So, \(\psi \) is a function from the set \({\mathcal {J}}\left( S\right) \) of the ideals of S,  to the set \(\mathcal { J}(R)\) of the ideals of R.

Lemma 1

The function \(\psi :{\mathcal {J}}(S)\rightarrow {\mathcal {J}}(R)\) has the following properties:

  1. (i)

    If P is a prime ideal of S then \(\psi (P)\) is a prime ideal of R not containing S.

  2. (ii)

    If P and Q are prime ideals of S such that \(\psi (P)=\psi (Q)\) then \(P=Q.\)

  3. (iii)

    If Q is a prime ideal of R not containing S then \(Q\cap S\) is a prime ideal of S and \(\psi (Q\cap S)=Q.\)

  4. (iv)

    \(\psi \left( \bigcap \limits _{P\in \textrm{Spec}S}P\right) =\bigcap \limits _{P\in \textrm{Spec}S}\psi (P).\)

Proposition 2

The function \(\psi :\textrm{Spec}S\rightarrow \textrm{Spec}R\) is injective, continuous and open onto its image.

Proof

By (iii) of the previous lemma we have that \(\psi \) is injective.

For continuity, it is enough to observe that if \(r\in R\) then \(\psi ^{-1}\left( D_{R}\left( r\right) \right) =\bigcup \nolimits _{s\in S}D_{S}\left( rs\right) .\)

On the other hand, if \(s\in S\) then it is easy to see that \(\psi \left( D_{S}\left( s\right) \right) =\psi \left( \textrm{Spec}S\right) \cap D_{R}\left( s\right) ,\) then \(\psi \) is open onto its image. \(\square \)

Hereinafter we denote with \(\textrm{Spec}_{S}R\) the image of the function \( \psi \) restricted to \(\textrm{Spec}S.\) In other words, \(\textrm{Spec}_{S}R\) is the set \(\{Q\in \textrm{Spec}R:Q\nsupseteq S\}.\) Thus, \(\textrm{Spec}S\) is homeomorphic to \(\textrm{Spec}_{S}R,\) seen as subspace of \(\textrm{Spec} R. \)

Proposition 3

\(\textrm{Spec}_{S}R\) is an open of \(\textrm{Spec}R\) and its closure is a subspace of \(\textrm{Spec}R\) homeomorphic to \(\textrm{Spec}\left( R/\psi \left( N\left( S\right) \right) \right) .\)

Proof

The first assertion follows from the equality \(\textrm{Spec}_{S}R=\bigcup \nolimits _{s\in S}D_{R}\left( s\right) .\) On the other hand:

$$\begin{aligned} \overline{\textrm{Spec}_{S}R}= & {} \left\{ Q\in \textrm{Spec}R:Q\supseteq \bigcap \limits _{P\in \textrm{Spec}S}\psi (P)\right\} \\= & {} \left\{ Q\in \textrm{Spec}R:Q\supseteq \psi \left( \bigcap \limits _{P\in \textrm{Spec}S}P\right) \right\} \\= & {} \{Q\in \textrm{Spec}R:Q\supseteq \psi \left( N\left( S\right) \right) \} \\= & {} V_{R}\left( \psi \left( N\left( S\right) \right) \right) \\\approx & {} \textrm{Spec}\left( R/\psi \left( N\left( S\right) \right) \right) . \end{aligned}$$

\(\square \)

Remark 4

Notice that the inclusion of \(\textrm{Spec}S\) in \(\textrm{Spec}\left( R/\psi \left( N\left( S\right) \right) \right) \) is given by the function

$$\begin{aligned} \lambda :\textrm{Spec}S\rightarrow \textrm{Spec}\left( R/\psi \left( N\left( S\right) \right) \right) :P\mapsto \psi (P)/\psi \left( N\left( S\right) \right) . \end{aligned}$$

Corollary 5

If R is spectrally compact (in particular if R has identity), then \( \textrm{Spec}\left( R/\psi \left( N\left( S\right) \right) \right) \) is a compactification of \(\textrm{Spec}S\) in which \(\textrm{Spec}S\) is open.

Definition 6

If R is spectrally compact, the space \(\textrm{Spec}\left( R/\psi \left( N\left( S\right) \right) \right) \) is called the R-nilcompactification of \( \textrm{Spec}S.\)

It is well known that every ring is an ideal of a unitary ring (see [8]), therefore we obtain the following result:

Theorem 7

The spectrum of every ring has a spectral compactification.

Proof

It is enough to observe that if S is a ring, we can choose R unitary and hence \(\textrm{Spec}\left( R/\psi \left( N\left( S\right) \right) \right) \) is a spectral space. \(\square \)

3 Functorial behavior of nilcompactification

Consider the category \({\mathcal {E}}\) whose objects are the pairs \(\left( S,R\right) \) with R a unitary i-extension of S and where the morphisms from \(\left( S_{1},R_{1}\right) \) to \(\left( S_{2},R_{2}\right) \) are the homomorphisms of unitary rings from \(R_{1}\) to \(R_{2}\) such that \( h(S_{1})=S_{2}.\)

As S and R vary, in this context, the functions \(\psi \) and \( \lambda \) defined in the previous section will be denoted by \(\psi _{\left( S,R\right) }\) and \(\lambda _{(S,R)}\) respectively.

For each object (S, R) of \({\mathcal {E}},\) we define \(Q(S,R)=R/\psi _{\left( S,R\right) }\left( N(S)\right) .\)

The following proposition can be proved without difficulty.

Proposition 8

If \(h:(S,R)\rightarrow (T,M)\) is a morphism of \( {\mathcal {E}}\), then

$$\begin{aligned} Q(h):Q(S,R)\rightarrow Q(T,M):r+\psi _{\left( S,R\right) }\left( N(S)\right) \mapsto h(r)+\psi _{\left( T,M\right) }\left( N(T)\right) \end{aligned}$$

is well defined and is a homomorphism of unitary rings.

Thus, Q is a functor from the category \({\mathcal {E}}\) to the category \( {{\mathcal {C}}}{{\mathcal {R}}}_{1}\) of unitary commutative rings.

We denote \(\textrm{NC}\) the contravariant functor \(\textrm{Spec}\circ Q:{\mathcal {E}}\rightarrow {\mathbb {S}}\) where \({\mathbb {S}}\) is the category of the spectral spaces and strongly continuous functions. So, \(\textrm{NC}(S,R)\) is the R-nilcompactification of \(\textrm{Spec}S.\)

Some natural transformations:

Consider the functors \(V:{\mathcal {E}}\rightarrow {{\mathcal {C}}}{{\mathcal {R}}}_{1}\) and \(W:{\mathcal {E}}\rightarrow {{\mathcal {C}}}{{\mathcal {R}}}^{s}\) defined as follows:

$$\begin{aligned}{} & {} V(h:(S,R)\rightarrow (T,M))=h:R\rightarrow M \hbox { and } \\{} & {} W(h:(S,R)\rightarrow (T,M))=h\mid _{S}:S\rightarrow T. \end{aligned}$$

The following proposition is a direct consequence of Proposition 8:

Proposition 9

For each object (S, R) of \({\mathcal {E}}\), denote by \(\theta _{(S,R)}:R\rightarrow Q(S,R)\) the canonical function to the quotient. Then \(\theta =\left( \theta _{\left( S,R\right) }\right) _{\left( S,R\right) \in Ob{\mathcal {E}}}\) is a natural transformation from the functor V to the functor Q.

figure a

Proposition 10

\(\psi =\left( \psi _{(S,R)}\right) _{(S,R)\in Ob{\mathcal {E}}}\) is a natural transformation from the functor \(\textrm{Spec}\circ W\) to the functor \( \textrm{Spec}\circ V.\)

Proof

It is enough to observe that if \(h:(S,R)\rightarrow (T,M)\) is a morphism of \({\mathcal {E}}\), then for each prime ideal P of T we have that \(h^{-1}\left( \psi _{(T,M)}\left( P\right) \right) =\psi _{(S,R)}\left( h^{-1}\left( P\right) \right) .\) \(\square \)

Proposition 11

\(\lambda =\left( \lambda _{(S,R)}\right) _{(S,R)\in Ob{\mathcal {E}}}\) is a natural transformation from the functor \( \textrm{Spec}\circ W\) to the functor \(\textrm{NC}.\)

figure b

Proof

Let \(h:(S,R)\rightarrow (T,M)\) be a morphism of \({\mathcal {E}}\), \(P\in \textrm{ Spec}T\) and \(r\in R.\)

$$\begin{aligned} \begin{array}{ll} r+\psi _{(S,R)}\left( N(S)\right) &{}\in Q(h)^{-1}\left( \psi _{(T,M)}\left( P\right) /\psi _{(T,M)}\left( N(S)\right) \right) \\ &{}\Leftrightarrow h(r)\in \psi _{(T,M)}\left( P\right) \\ &{}\Leftrightarrow h(r)T\subseteq P \\ &{}\Leftrightarrow h(rS)\subseteq P \\ &{}\Leftrightarrow rS\subseteq h^{-1}(P) \\ &{}\Leftrightarrow r\in \psi _{(S,R)}\left( h^{-1}(P)\right) \\ &{}\Leftrightarrow r+\psi _{(S,R)}\left( N(S)\right) \in \psi _{(S,R)}\left( h^{-1}(P)\right) /\psi _{(S,R)}\left( N(S)\right) \end{array} \end{aligned}$$

Thus, \(NC(h)\left( \lambda _{(T,M)}\left( P\right) \right) =\lambda _{(S,R)}\left( \left( \textrm{Spec}\circ W\right) \left( h\right) \left( P\right) \right) .\) \(\square \)

Proposition 12

Let \(h:(S,R)\rightarrow (T,M)\) be a morphism of the category \({\mathcal {E}}.\) The function h can be restricted to a function from \(\psi _{\left( S,R\right) }\left( N(S)\right) \) to \(\psi _{(T,M)}\left( N(T)\right) .\)

Proof

Consider \(r\in \psi _{\left( S,R\right) }\left( N(S)\right) ,\) that is, \(rs\in N(S)\) for all \(s\in S.\) Given \(t\in T,\) there exists \(s\in S\) such that \(h\left( s\right) =t.\) Thus, \(h\left( r\right) t=h\left( r\right) h\left( s\right) =h\left( rs\right) \in h\left( N\left( S\right) \right) \subseteq N\left( T\right) \) so that \(h\left( r\right) \in \psi _{(T,M)}\left( N(T)\right) .\) \(\square \)

Let \(\chi :{\mathcal {E}}\rightarrow {{\mathcal {C}}}{{\mathcal {R}}}\) be the functor defined by:

$$\begin{aligned} \chi \left( h:(S,R)\rightarrow (T,M)\right) =h\mid _{\psi _{\left( S,R\right) }\left( N(S)\right) }:\psi _{\left( S,R\right) }\left( N(S)\right) \rightarrow \psi _{(T,M)}\left( N(T)\right) \end{aligned}$$

and for the object (S, R) of \({\mathcal {E}}\), denote by \(j_{(S,R)}\) the natural inclusion of \(\psi _{\left( S,R\right) }\left( N(S)\right) \) into the ring R.

Notice that for each morphism \(h:(S,R)\rightarrow (T,M)\) in the category \({\mathcal {E}}\), the square in the following figure is commutative; this allows us to state the following result.

figure c

Proposition 13

\(j=\left( j_{(S,R)}\right) _{(S,R)\in Ob{\mathcal {E}}}\) is a natural transformation from the functor \(\chi \) to the functor V.

4 First variation

Fix a ring S. Let \({\mathcal {E}}\left( S\right) \) be the subcategory of \( {{\mathcal {C}}}{{\mathcal {R}}}_{1}\) whose objects are the i-extensions of S and whose morphisms are those that can be restricted to the identity of S. It is clear that \({\mathcal {E}}(S)\) can be identified with a subcategory of \( {\mathcal {E}}\) and therefore the functor Q can be restricted to a functor \( Q_{S}:{\mathcal {E}}(S)\rightarrow {{\mathcal {C}}}{{\mathcal {R}}}_{1}\) and the functor \(\textrm{NC }\) can be restricted to a functor \(\textrm{NC}_{S}:{\mathcal {E}}(S)\rightarrow {\mathbb {S}}.\)

Proposition 14

If \(h:R\rightarrow T\) is a morphism of \({\mathcal {E}}(S)\), then \( Q_{S}(h):Q(S,R)\rightarrow Q(S,T)\) is injective.

Proof

$$\begin{aligned} r+\psi _{(S,R)}\left( N(S)\right)\in & {} \ker Q_{S}(h)\Leftrightarrow h(r)\in \psi _{(S,T)}\left( N(S)\right) \\\Leftrightarrow & {} h(r)s\in N(S)\text { for all }s\in S \\\Leftrightarrow & {} h(rs)\in N(S)\text { for all }s\in S \\\Leftrightarrow & {} rs\in N(S)\text { for all }s\in S \\\Leftrightarrow & {} r\in \psi _{(S,R)}\left( N(S)\right) \\\Leftrightarrow & {} r+\psi _{(S,R)}\left( N(S)\right) =0 \end{aligned}$$

\(\square \)

Corollary 15

If \(h:R\rightarrow T\) is a surjective morphism of \({\mathcal {E}}(S)\), then \(Q_{S}(h):Q(S,R)\rightarrow Q(S,T)\) is an isomorphism and therefore \(\textrm{NC}_{S}(h):\textrm{NC}_{S}\left( T\right) \rightarrow \textrm{NC}_{S}\left( R\right) \) is a homeomorphism.

Corollary 16

If \(h:R\rightarrow T\) is a morphism of \({\mathcal {E}}(S)\), then \(\textrm{NC} _{S}(h)\left( \textrm{NC}_{S}\left( T\right) \right) \) is dense in \(\textrm{ NC}_{S}\left( R\right) .\)

Proof

It is consequence of the injectivity of \(Q_{S}(h):Q_{S}\left( R\right) \rightarrow Q_{S}(T)\) (see [5]). \(\square \)

Proposition 17

If \(h:R\rightarrow T\) is a morphism of \({\mathcal {E}}(S)\), then \(\textrm{NC} _{S}(h)\circ \lambda _{(S,T)}=\lambda _{(S,R)}.\)

figure d

The proof is a direct consequence of Proposition 11.

Corollary 18

If \(h:R\rightarrow T\) is a morphism of \({\mathcal {E}}(S)\), then \(\textrm{Spec}S\) is a subspace of \(\textrm{NC}_{S}(h)\left( \textrm{NC}_{S}\left( T\right) \right) \!.\)

5 Second variation

Given a ring S, we denote by \(U_{0}\left( S\right) \) the set \(S\times \mathbb { Z}\) endowed with the operations:

$$\begin{aligned} (s,\alpha )+(t,\beta )= & {} (s+t,\alpha +\beta ) \\ (s,\alpha )(t,\beta )= & {} (st+\beta s+\alpha t,\alpha \beta ). \end{aligned}$$

It is well known that \(U_{0}\left( S\right) \) is a unitary ring and that, if we identify S with \(S_{0}=S\times \{0\},\) S is an ideal of \(U_{0}(S).\) Besides we have the following universal property:

Theorem 19

Let S be a ring and consider \(\iota _{S}:S\rightarrow U_{0}(S):s\mapsto (s,0).\) If R is a unitary ring and \(g:S\rightarrow R\) is a homomorphism, then there exists a unique homomorphism of unitary rings \({\widetilde{g}} :U_{0}(S)\rightarrow R\) such that \({\widetilde{g}}\circ \iota _{S}=g.\)

figure e

This property allows us to extend \(U_{0}\) to a functor from the category \( {{\mathcal {C}}}{{\mathcal {R}}}\) of commutative rings to the category \({{\mathcal {C}}}{{\mathcal {R}}}_{1}\) of unitary rings, defining \(U_{0}(h)=\widetilde{\iota _{T}\circ h}\), for a homomorphism \(h:S\rightarrow T\). We have also that \(\iota =(\iota _{S})_{S\in Ob{{\mathcal {C}}}{{\mathcal {R}}}}\) is a natural transformation from the identity functor of the category \({{\mathcal {C}}}{{\mathcal {R}}}\) to the functor \(U_{0}\) considered as endo-functor of \({{\mathcal {C}}}{{\mathcal {R}}}.\)

Notice that if \(h:S\rightarrow T\) is a surjective homomorphism then \( U_{0}(h) \) is a morphism in the category \({\mathcal {E}}\) from the object \( (S,U_{0}(S))\) to the object \((T,U_{0}(T)).\) Thus, \(U_{0}\) can be seen as a functor from the category \({{\mathcal {C}}}{{\mathcal {R}}}^{s}\) of commutative rings and surjective homomorphisms to the category \({\mathcal {E}}.\)

figure f

The universal property of \(U_{0}\) allows us to conclude immediately the following theorem:

Theorem 20

For each ring S,  \(U_{0}(S)\) is an initial object of the category \(\mathcal { E}(S).\)

Proof

Let R be an object of \({\mathcal {E}}(S)\) and let \(u:S\rightarrow R\) be the inclusion homomorphism. The universal property guarantees that there is a unique homomorphism of unitary rings \(u_{R}:U_{0}(S)\rightarrow R\) such that \(u_{R}\circ \iota _{S}=u,\) that is, \(u_{R}\) restricted to S is the identity. \(\square \)

The image of \(\textrm{NC}_{S}\) is a subcategory of \({\mathbb {S}}\), that we denote by \({{\mathcal {N}}}{{\mathcal {C}}}(S)\) (category of nilcompactifications of S). We denote by \(\textrm{NC}_{0}\) the functor \(\textrm{NC}\circ U_{0}:{{\mathcal {C}}}{{\mathcal {R}}}^{s}\rightarrow {\mathbb {S}}.\) Thus, we have the following result:

Corollary 21

For each ring S,  \(\textrm{NC}_{0}\left( S\right) \) is a final object of the category \({{\mathcal {N}}}{{\mathcal {C}}}(S).\)

Remark 22

As a consequence of Corollary 15 we have that \( \textrm{NC}_{0}\left( S\right) \) is homeomorphic to \(\textrm{NC}_{0}\left( S/N(S)\right) .\) Therefore we can reduce the study of nilcompactifications to semi-prime or reduced rings.

6 Third variation

In this section we are working again with a fixed ring S. Let R be an object of \({\mathcal {E}}(S)\) and consider the unique morphism \( u_{R}:U_{0}(S)\rightarrow R\) of \({\mathcal {E}}(S).\) Then \(\textrm{NC} _{S}(u_{R}):\textrm{NC}_{S}(R)\rightarrow \textrm{NC}_{0}\left( S\right) \) is a (strongly) continuous function. We denote by \(\eta (R)\) the image of the function \(\textrm{NC}_{S}(u_{R}).\)

Theorem 23

If R is an object of \({\mathcal {E}}(S)\), then \(\eta (R)\) is a compactification of \(\textrm{Spec}S\) where \(\textrm{Spec}S\) is open, and besides it is dense in \(\textrm{NC}_{0}\left( S\right) .\)

Proof

It is enough to observe that \(\textrm{Spec}S\subseteq \eta (R)\subseteq \textrm{NC} _{0}\left( S\right) \) and that \(\textrm{Spec}S\) is an open dense subspace of \(\textrm{ NC}_{0}\left( S\right) .\) \(\square \)

We consider the pre-order between compactifications given in [11] : let \(\left( X^{\prime },\tau ^{\prime }\right) \) and \(\left( X^{\prime \prime },\tau ^{\prime \prime }\right) \) be compactifications of \(\left( X,\tau \right) ,\) with immersions \(f^{\prime }:\left( X,\tau \right) \rightarrow \left( X^{\prime },\tau ^{\prime }\right) \) and \(f^{\prime \prime }:\left( X,\tau \right) \rightarrow \left( X^{\prime \prime },\tau ^{\prime \prime }\right) \). Set \(\left( X^{\prime },\tau ^{\prime }\right) \) \(\preceq \left( X^{\prime \prime },\tau ^{\prime \prime }\right) \) if there exists \(h:\left( X^{\prime \prime },\tau ^{\prime \prime }\right) \longrightarrow \left( X^{\prime },\tau ^{\prime }\right) \) continuous and surjective, such that \(f^{\prime }=h\circ f^{\prime \prime }\) . We immediately obtain the following result:

Proposition 24

If R is an object of \({\mathcal {E}}(S)\), then \(\eta (R)\) is a compactification of \(\textrm{Spec}S\) smaller than \(\textrm{NC}_{S}(R).\)

Corollary 25

If \(\textrm{NC}_{0}\left( S\right) \) is a Hausdorff space, then it is the smallest nilcompactification of \(\textrm{Spec}S.\)

Proof

If R is an object of \({\mathcal {E}}(S)\) then \(\eta (R)\) is a compact subset of \(\textrm{NC}_{0}\left( S\right) \), then it is closed. Besides, \(\eta (R)\) is dense in \(\textrm{NC}_{0}\left( S\right) \), therefore \(\textrm{NC}_{S}(u_{R})\) is surjective. \(\square \)

The proof of the following proposition is a simple routinary exercise:

Proposition 26

If \(h:R\rightarrow T\) is a morphism of the category \({\mathcal {E}}(S)\), then \( \eta (T)\subseteq \eta (R).\)

If for each morphism \(h:R\rightarrow T\) of the category \({\mathcal {E}}(S)\) we define \(\eta (h):\eta (T)\rightarrow \eta (R):x\mapsto x\), then \(\eta \) is a functor from the category \({\mathcal {E}}(S)\) to the category \(\textrm{Top}\) of topological spaces and continuous functions.

We have then the following natural question: if R is an object of \( {\mathcal {E}}(S)\), is \(\eta (R)\) a spectral space?

7 Two examples

In this section we present two examples that illustrate some results of this paper.

Example 27

Let B be a Boolean ring without identity. As B is semiprime, \(N\left( B\right) =0\). Let us compute \(\psi _{\left( B,U_{0}\left( B\right) \right) }\left( 0\right) :\)

\(\left( a,\alpha \right) \in \psi _{\left( B,U_{0}\left( B\right) \right) }\left( 0\right) \) if and only if \(ax+\alpha x=0\) for each \(x\in B,\) that is, \(ax=\alpha x\) for each \(x\in B.\)

If \(\alpha \) is even then \(ax=0\) for all \(x\in B,\) from which it follows that \(a=0.\)

If \(\alpha \) is odd then \(ax=x\) for all \(x\in B\) and therefore, a is the identity of B,  which is a contradiction.

Hence, \(\psi _{\left( B,U_{0}\left( B\right) \right) }\left( 0\right) =\left\{ 0\right\} \times 2{\mathbb {Z}}\) and \(Q\left( B,U_{0}\left( B\right) \right) =U_{0}\left( B\right) /\left( \left\{ 0\right\} \times 2 {\mathbb {Z}}\right) .\)

In this case, \(NC\left( B,U_{0}\left( B\right) \right) \) is precisely the Alexandroff compactification of \(\textrm{Spec}B\) (see [1]).

The following example shows that different i-extensions of a ring can produce different nilcompactifications:

Example 28

Consider the ring without identity \(S=x{\mathbb {R}}\left[ x\right] \). Two different unitary i-extensions of S are \(U_{0}\left( S\right) \) and \( {\mathbb {R}}\left[ x\right] .\) Notice that the unique homomorphism from \( U_{0}\left( S\right) \) to \({\mathbb {R}}\left[ x\right] \) is not surjective and there not exists a homomorphism from \({\mathbb {R}}\left[ x\right] \) to \( U_{0}\left( S\right) .\) It is easy to see that \(\psi _{\left( S,U_{0}\left( S\right) \right) }\left( 0\right) =0\) and \(\psi _{\left( S,{\mathbb {R}}\left[ x \right] \right) }\left( 0\right) =0.\) Therefore, \(Q\left( S,U_{0}\left( S\right) \right) =U_{0}\left( S\right) \) and \(Q\left( S,{\mathbb {R}}\left[ x \right] \right) ={\mathbb {R}}\left[ x\right] ,\) thus \(NC\left( S,U_{0}\left( S\right) \right) =\textrm{Spec}\left( U_{0}\left( S\right) \right) \) and \( NC\left( S,{\mathbb {R}}\left[ x\right] \right) =\textrm{Spec}{\mathbb {R}}\left[ x \right] \). If we consider the surjective homomorphisms

$$\begin{aligned}{} & {} \pi :U_{0}\left( S\right) \rightarrow {\mathbb {Z}}:\left( s,z\right) \mapsto z, \\{} & {} \beta :{\mathbb {R}}\left[ x\right] \rightarrow {\mathbb {R}}:p\left( x\right) \mapsto p\left( 0\right) , \end{aligned}$$

we conclude, by the Correspondence Theorem, that \(NC\left( S,U_{0}\left( S\right) \right) \) is a compactification of \(\textrm{Spec}S\) by enumerable points, while \(NC\left( S,{\mathbb {R}}\left[ x\right] \right) \) is a compactification of \(\textrm{Spec}S\) by one point.

The following diagram summarizes the ideas presented in this paper.

figure g