On Nilcompactifications of Prime Spectra of Commutative Rings

Given a ring R and S one of its proper ideals, we obtain a compactification of the prime spectrum of S through a mainly algebraic process. We name it the R-nilcompactification of SpecS. We study some categorical properties of this construction.

Compactification of a topological space is an important topic considered in a wide range of branches in mathematics as it guarantees a useful property for the space, see for example [4], [9], [10], [11], [12], [13], [14].In this paper we study, from a categorical point of view, the compactification method of prime spectra which is presented in [3], named nilcompactification.Nilcompactification is a topological method obtained mainly through an algebraic process.The categorical point of view of nilcompactification, with some of its possible variations, offers an interesting wealth for this process.For the categorical concepts see, for example, [8].This method of nilcompactification is functorial in a simple way in that it has interesting properties and the involved constructions provide us with different natural transformations.We can take into account that some compactification processes need to consider suitable subcategories to obtain a functorial behavior, as studied in [2] for the Alexandroff compactifications.
In 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 SpecS and it is the set of prime ideals of S endowed with the Zariski topology.In this topology the sets D S (a) = {P ∈ SpecS : a / ∈ P }, where a ∈ S, provide a basis.The closed sets are 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) → Spec (S/I) : P → P/I is a homeomorphism (see [5]).The Zariski topology is also called the hull-kernel topology because the closure of a subset B of SpecS is the set The nilradical of S, denoted N (S) , is precisely the kernel of SpecS.
It is known that S is semiprime or reduced if N (S) = {0} .
A ring whose spectrum is compact is a spectrally compact ring.In particular, every unitary ring R is spectrally compact because D R (1) = SpecR.
We use the following notations: CR : Category of commutative rings and homomorphisms of rings.CR 1 : Category of unitary commutative rings and homomorphisms of unitary rings.CR s : Category of commutative rings and surjective homomorphisms.Top : Category of topological spaces and continuous functions.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 [6]).A function is strongly continuous if it sends compact open sets in compact open sets by reciprocal image.

The mechanism of 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 ψ(I) = {x ∈ R : xS ⊆ I} is an ideal of R. So, ψ is a function from the set J (S) of the ideals of S, to the set J (R) of the ideals of R.
Lemma 1.The function ψ : J (S) → J (R) has the following properties: (i) If P is a prime ideal of S then ψ(P ) is a prime ideal of R not containing S.
(ii) If P and Q are prime ideals of S such that ψ(P Proof.By (iii) of the previous lemma we have that ψ is injective.
For continuity, it is enough to observe that if r ∈ R then On the other hand, if s ∈ S then it is easy to see that ψ Hereinafter we denote with Spec S R the image of the function ψ restricted to SpecS.In other words, Spec S R is the set {Q ∈ SpecR : Q S}.Thus, SpecS is homeomorphic to Spec S R, seen as subspace of SpecR.Proposition 3. Spec S R is an open of SpecR and its closure is a subspace of SpecR homeomorphic to Spec (R/ψ (N (S))) .

Proof. The first assertion follows from the equality Spec
On the other hand: Remark 4. Notice that the inclusion of SpecS in Spec (R/ψ (N (S))) is given by the function λ : SpecS → Spec (R/ψ (N (S))) : P → ψ(P )/ψ (N (S)) .
It is well known that every ring is an ideal of a unitary ring (see [7]), 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 Spec (R/ψ (N (S))) is a spectral space.

Functorial behavior of the mechanism of nilcompactification
Consider the category E whose objects are the pairs (S, R) with R a unitary i-extension of S and where the morphisms from (S 1 , R 1 ) to (S 2 , R 2 ) are the homomorphisms of unitary rings from R 1 to R 2 such that h(S 1 ) = S 2 .
As S and R are variables, in this context, the functions ψ and λ defined in the previous section will be denoted ψ (S,R) and λ (S,R) respectively.
The following proposition can be proved without difficulty.
is well defined and is a homomorphism of unitary rings.
Thus, Q is a functor from the category E to the category CR 1 of unitary commutative rings.
We denote NC the contravariant functor Spec • Q : E → S where S is the category of the spectral spaces and the strongly continuous functions.So, NC(S, R) is the R-nilcompactification of SpecS.

Some natural transformations:
Consider the functors V : E → CR 1 and W : E → CR s defined as follows: The following proposition is a direct consequence of the Proposition 8: Proposition 9.If for each object (S, R) of E, we denote by θ (S,R) : R → Q(S, R) the canonical function to the quotient, then θ = θ (S,R) (S,R)∈ObE is a natural transformation from the functor V to the functor Q.
Proposition 10. ψ = ψ (S,R) (S,R)∈ObE is a natural transformation from the functor Spec • W to the functor Spec • V.
Proof.It is enough to observe that if h : (S, R) → (T, M ) is a morphism of E then for each prime ideal P of T we have that h −1 ψ (T,M ) (P ) = ψ (S,R) h −1 (P ) .
Proposition 11. λ = λ (S,R) (S,R)∈ObE is a natural transformation from the functor Spec • W to the functor NC.
Proof.Let h : (S, R) → (T, M ) be a morphism of E, P ∈ SpecT and r ∈ R.
Proposition 12. Let h : (S, R) → (T, M ) be a morphism of the category E. The function h can be restricted to a function from ψ (S,R) (N (S)) to ψ (T,M ) (N (T )) .
Proof.Consider r ∈ ψ (S,R) (N (S)) , that is, rs ∈ N (S) for all s ∈ S. Given t ∈ T, there exists s ∈ S such that h (s) = t.Thus, h (r Let χ : E → CR be the functor defined by: and for the object (S, R) of E, denote j (S,R) the natural inclusion of ψ (S,R) (N (S)) into the ring R.
Notice that for each h : (S, R) → (T, M ), morphism of the category E, the square in the following figure is commutative; this allows us to state the following result.
Proposition 13. j = j (S,R) (S,R)∈ObE is a natural transformation from the functor χ to the functor V.

First variation
Fix a ring S. Let E (S) be the subcategory of CR 1 whose objects are the iextensions of S and whose morphisms are those that can be restricted to the identity of S. It is clear that E(S) can be identified with a subcategory of E and therefore the functor Q can be restricted to a functor Q S : E(S) → CR 1 and the functor NC can be restricted to a functor NC S : E(S) → S. Proof.It is consequence of the injectivity of Q S (h) : Q S (R) → Q S (T ) (see [5]).
Proof.It is a direct consequence of the Proposition 11.
Corollary 18.If h : R → T is a morphism of E(S) then SpecS is a subspace of NC S (h) (NC S (T )) .

Second variation
Given a ring S, we denote U 0 (S) the set S × Z endowed with the operations: It is well known that U 0 (S) is a unitary ring and that, if we identify S with S 0 = S × {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 ι S : S → U 0 (S) : s → (s, 0).If R is a unitary ring and g : S → R is a homomorphism, then there exists a unique homomorphism of unitary rings g :

R
This property allows us to extend U 0 to a functor from the category CR of commutative rings to the category CR 1 of unitary rings, defining U 0 (h) = ι T • h, for a homomorphism h : S → T. We have also that ι = (ι S ) S∈ObCR is a natural transformation from the identity functor of the category CR to the functor U 0 considered as endo-functor of CR.
Notice that if h : S → T is a surjective homomorphism then U 0 (h) is a morphism in the category 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 CR s of commutative rings and surjective homomorphisms to the category E.
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 E(S).

Proposition 2 .
The function ψ : SpecS → SpecR is injective, continuous and open onto its image.
R) (N (S)) = 0 Corollary 15.If h : R → T is a surjective morphism of E(S) then Q S (h) : Q(S, R) → Q(S, T ) is an isomorphism and therefore NC S (h) : NC S (T ) → NC S (R) is a homeomorphism.Corollary 16.If h : R → T is a morphism of E(S) then NC S (h) (NC S (T )) is dense in NC S (R) .

Example 28 .
Consider the ring without identity S = xR [x].Two different unitary i-extensions of S are U 0 (S) and R [x].Notice that the unique homomorphism from U 0 (S) to R [x] is not surjective and there not exists a homomorphism from R [x] to U 0 (S) .It is easy to see that ψ (S,U 0 (S)) (0) = 0 andψ (S,R[x]) (0) = 0. Therefore, Q (S, U 0 (S)) = U 0 (S) and Q (S, R [x]) R [x] , thus N C (S, U 0 (S)) = Spec (U 0 (S)) and N C (S, R [x]) = SpecR [x].If we consider the surjective homomorphisms π : U 0 → Z : (s, z) → z, β : R [x] → R : p (x) → p (0) ,we conclude, by the Correspondence Theorem, that N C (S, U 0 (S)) is a compactification of SpecS by enumerable points, while N C (S, R [x]) is a compactification of SpecS by one point.The following diagram summarizes the ideas presented in this paper.