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Echi Othman, Adel Khalfallah On the prime spectrum of the ring of bounded nonstandard complex numbers Proceedings of the American Mathematical Society DOI: 10.1090/proc/14204 Accepted Manuscript

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The Version of Record is accessible to everyone ﬁve years after publication in an issue. ON THE PRIME SPECTRUM OF THE RING OF BOUNDED NONSTANDARD COMPLEX NUMBERS

OTHMAN ECHI AND ADEL KHALFALLAH

Abstract. In this paper, we provide some algebraic structures of convex subrings of ∗C, a nonstandard extension of the ﬁeld of complex numbers C. In particular, a detailed description of the prime spectrum of any convex subring of ∗C is given. To achieve our goal, ﬁrst we investigate prime ideals and we characterize two consecutive elements in the spectrum of a divided domain. We also show that the prime spectrum of the ring of bounded hypercomplex numbers has two peculiar properties: there are no three consecutive elements in the spectrum, moreover, non-zero elements are disjoint union of three subsets one of them is strongly dense and the two others are dense in the spectrum.

1. Introduction Historically, the idea of nonstandard analysis was to rigorously justify calculations with infinitesimal numbers. Infinitesimals played a fundamental role in the development of the calculus with the work of Newton and Leibniz. Nowadays, nonstandard analysis has gone far beyond the realm of infinitesimals. In fact, it provides a machinery which enables one to describe explicitly mathematical concepts which by standard methods can only be described implicitly and in a cumbersome way. In 1960, Abraham Robinson gave foundations for the use of infinitesimals in all the branches of mathematics, see [22]. Robinson’s foundation started with the real number system. In nonstandard analysis, the standard part mapping is a function from bR, the bounded (finite) hyperreal numbers, to the real numbers. This concept plays a key role in defining the concepts of the calculus.

The main objective of this paper is to provide a detailed description of the prime spectrum of bC (resp. bR), the ring of bounded hypercomplex (resp. hyperreal) numbers and more generally convex subrings of ∗C (resp. ∗R). We should mention that convex subrings of ∗C (resp. ∗R) are simply subrings of ∗C (resp. ∗R) containing bC (resp. bR).

In [15], bC is the ground ring of some bounded polynomial algebras. These algebras are the local models to the category of bC-bounded schemes. And, the authors constructed a functor, called the standard part functor, from the category of bC-bounded schemes to the

2010 Mathematics Subject Classiﬁcation. Primary: 26E35, 03H05, 13A15, Secondary: 13F30, 12J20, 12J25. Key words and phrases. Nonstandard analysis, Divided domains, Valuation domains, Prime spectrum. 1

This16 isApr a pre-publication 2018 16:25:53 version of thisEDT article, which may differ from the final published version. CopyrightAlgebra+NT+Comb+Logi restrictions may apply. Version 2 - Submitted to Proc. Amer. Math. Soc. 2 OTHMAN ECHI AND ADEL KHALFALLAH

category of analytic complex spaces.

In [24], Todorov introduced convex rings to develop the nonstandard version of the Colombeau’s theory of generalized functions, see [20, 25]. This nonstandard approach provides significant improvements in the set of scalars of functional spaces. The scalar of the nonstandard version of Colombeau’s theory is given by the factor field Fb = F/iF, where F is a convex subring of ∗R and iF denotes its maximal ideal. The factor ring Fb is a totally ordered real closed non-archimedean field extension of R while its standard counterpart in the classical theory of Colombeau’s generalized functions is a partially ordered nonarchimedean ring containing R with zero divisors.

In [16, 17], the authors developed the nonstandard counterpart of the space of Colombeau’s holomorphic generalized functions using convex subrings of ∗C. The need to go in such ∗ theory to other convex subrings of C than only Mρ (see Examples 3.2.1 (iii)) has been apparently in many applications and has inspired other authors to use subrings deﬁned by asymptotic scales [7].

Aschenbrenner and Goldbring [2] studied the relationship between fields of transseries and residue fields of convex subrings of ∗R. They proved that the field of exponential- logarithmic series (or EL-series), first defined in [12], is embedded in the exponential field Ecρ, see Examples 3.2.1 (v). Transseries naturally arise in solving differential equations at infinity and studying the asymptotic behavior of their solutions, where ordinary power series, Laurent series are inadequate. For a comprehensive introduction to transseries and asymptotic differential algebra, the reader is refereed to [3]. Some of the rings and fields studied in [3] are isomorphic to (or can be embedded in) convex rings and residue fields discussed in our paper.

These developments make it clear that it is important to study the algebraic and the topological properties of convex subrings of ∗C.

In order to reach our goal, ﬁrst we investigate divided rings. Let us recall that a commutative integral domain R is said to be divided in case each prime ideal p of R is divided;

that is p = pRp . An important class of divided integral domains is provided by valuations domains. It is worth noting that these domains were studied by Akiba as AV-domains [1].

We have included enough material and motivation in order to make our paper as self- contained as possible.

The paper is organized as follows: in the second section we recall the deﬁnition of divided rings and characterizations of principal quotient rings which is the convenient setting to study when a prime ideal has an immediate successor in a domain with linearly ordered spectrum. Next, we give a complete description of the spectrum of a divided domain.

The last section is devoted to the study of the prime spectrum of bC, the ring of bounded hypercomplex numbers. Some properties can be deduced directly from the description of the spectrum of divided rings. We show that the height and the coheight of any nonzero, non-maximal prime ideal is inﬁnite. Moreover, we prove that the zero ideal has no immediate successor, and the maximal ideal has no immediate predecessor. Finally, we show that the spectrum of bC enjoys two peculiar properties: it is a disjoint union of three subsets one of them is strongly dense and the two others are dense in the spectrum; there are no three consecutive elements in the spectrum.

Throughout this paper “⊂” stands for proper containment and “⊆” for large containment.

2. Divided Domains The aim of this section is to give a description of the prime spectrum of a divided ring, see Theorem 2.3 and to provide a characterization of two consecutive elements in the prime spectrum of a divided ring, see Theorem 2.9. Some of these results might be known to experts but since no references can be found, we included short proofs.

First, let us recall the deﬁnition of divided rings. Deﬁnition 2.1. [8] Let R be a ring, p ∈ Spec(R) is called a divided prime ideal in R if p is comparable under inclusion to each (principal) ideal of R; and R is a divided ring if each p ∈ Spec(R) is divided in R. Clearly any valuation domain is divided.

The following proposition gives a characterization of divided domains. Proposition 2.2. [4, Proposition 2] The following statement are equivalent for an integral domain R (1) R is a divided domain. (2) For every a, b ∈ R, either a | b or b | an, for some n ≥ 1.

For a nonzero and nonunit element ρ of a domain R, we denote by \ n Iρ := ρ R. n≥1 2.1. The prime spectrum of a divided ring. The following theorem gives the structure of prime ideals of a divided domain. Theorem 2.3 (The Prime Spectrum of Divided Domains). Let R be a divided domain. Then Spec(R) is linearly ordered with maximal element m and ( ) \ Spec(R) = {m} ∪ Iρ : T is a nonempty subset of m \{0} . ρ∈T

The proof is based on the following lemmas Lemma 2.4. Let p be a non-maximal divided prime ideal of a quasi-local domain (R, m). Then \ p = Iρ. ρ∈m\p Proof. As p is a divided prime ideal; then it is comparable to any principal ideal of R. So, n if ρ ∈ m \ p, we have p ⊆ ρ R, for all n ≥ 1. This leads to p ⊆ Iρ; and consequently \ \ p ⊆ Iρ. Conversely, if x ∈ Iρ and x∈ / p, then x ∈ Ix, a contradiction. ρ∈m\p ρ∈m\p

Lemma 2.5. Let R be a divided domain with maximal ideal m, then Iρ is a prime ideal of R for every ρ ∈ m \{0}.

Proof. Assume that Iρ is not prime. Then there exist x, y ∈ R such that xy ∈ Iρ and n m x, y 6∈ Iρ. So there exist m, n ≥ 1, such that ρ - x and ρ - y. By Proposition 2.2, n n1 n m1 nn1 mm1 x | (ρ ) and x | (ρ ) for some n1, m1 ≥ 1. Thus ρ = xx1 and ρ = yy1 for some x1, y1 ∈ R. If follows that nn1+mm1 ρ = xyx1y1 ∈ Iρ, which is impossible as Iρ does not contains any power of ρ. Proof of Theorem 2.3. According to Lemma 2.4, we have ( ) \ Spec(R) ⊆ {m} ∪ Iρ : T is a nonempty subset of m \{0} . ρ∈T

Conversely, as in a divided domain R, every Iρ is prime (see Lemma 2.5) and Spec(R) is linearly ordered, we deduce, using [14, Theorem 9, page 6], that any intersection of a family of Iρ is a prime ideal of R. 2.2. Consecutive elements in the spectrum of a divided ring. To investigate when a prime ideal of a ring R has an immediate successor, we need to introduce the notion of principal quotient ring. An overring T of a domain R is called a principal quotient ring (pqr) if there exists ρ ∈ R \{0} such that T is a quotient ring of R with respect to the multiplicative set 2 n −1 S = {1, ρ, ρ , . . . , ρ ,...}; that is to say T = S R. In this case, T will be denoted by Rρ. We start by recalling a characterization when Rp, the localization at a prime ideal p, is pqr. Lemma 2.6. [21, Proposition 2.1] Let R be a domain, p be a prime ideal of R and ρ ∈ R\p. Then the following statements are equivalent. (1) Rp = Rρ. √ (2) For each b ∈ R \ p, ρ ∈ bR. (3) If q ∈ Spec(R) such that q * p, then ρ ∈ q.

(4) D(ρ) = (↓ p). where D(ρ) = {q ∈ Spec(R): ρ 6∈ q} and (↓ p) = {q ∈ Spec(R): q ⊂ p}.

The following lemma which is extracted from [5, Theorem 2.5] is essentially a direct consequence of Lemma 2.6. Lemma 2.7. Let R be a domain with linearly ordered prime spectrum, p be a non maximal prime ideal of R. Then the following statements are equivalent.

(1) Rp is pqr. (2)( ↓ p) is Zariski open in Spec(R). (3) p has an immediate successor (i.e., a prime ideal p1 of R such that p ⊂ p1 are adjacent in Spec(R)). In a domain with linearly ordered prime spectrum, divided prime ideals with immediate successor are necessarily of the form Iρ. Proposition 2.8. [9] Let R be a domain with linearly prime spectrum and maximal ideal m. Let p be a prime ideal of R having an immediate successor. Let ρ ∈ m \ p such that (↓ p) = D(ρ). Then the following properties hold.

(1) Iρ ⊆ p. (2) p = Iρ if and only if p is divided. Our second result in this section is the following Theorem 2.9. Let R be a divided domain with maximal ideal m and p ⊂ q be prime ideals of R. Then the following statements are equivalent. (1) p ⊂ q are consecutive. √ (2) There exists ρ ∈ m \{0} such that p = Iρ and q = ρR.

Proof. (1) =⇒ (2): As p has an immediate successor, then by Lemmas 2.6 and 2.7,√ there exists ρ ∈ m\{0} such that D(ρ) = (↓ p). By proposition 2.8, we get p = Iρ and q = ρR, which is the smallest prime ideal containing ρ. √ (2) =⇒ (1): Let r be a prime ideal such that Iρ ( r ⊆ ρR√. By Proposition 2.8, we have Iρ = ∪[q ∈ Spec(R): ρ∈ / q]. Hence r contains ρ, thus r = ρR. Recall that an ideal p of a domain R is said to be a G-ideal if {p} is locally closed in Spec(R). The set of all G-ideals of R is denoted by Gold(R). It is well known that Gold(R) is strongly dense in Spec(R), that is, it meets any non empty locally closed subset of Spec(R). Remark 2.10. If (R, m) is a divided domain, then Spec(R) is linearly ordered. Thus p is G-ideal if and only if (↓ p) is open, see [9]. Hence

Gold(R) = {Iρ : ρ ∈ m \{0}} ∪ {m}.

3. The Prime Spectrum of bC Firstly, let us recall preliminary notions providing a background necessary for the com- prehension of this section. 3.1. Nonstandard Analysis. The approach to nonstandard analysis that we use in the present paper follows that of Stroyan and Luxemburg [23]. For a comprehensive introduction to nonstandard analysis, the reader is referred to [11, 13, 19, 22]. S One starts with a superstructure V (C) = Vn(C) over C. The natural embedding ∗ : V (C) → V (∗C) satisﬁes : The Transfer Principle. Let Φ(x1, x2, . . . , xn) be a bounded formula of the superstructure V (R) and let A1,A2,...,An be elements of the superstructure V (C). Then the assertion Φ(A1,A2,...,An) about elements of V (C) holds true if and only if the ∗ ∗ ∗ ∗ assertion Φ( A1, A2,..., An) about elements of V ( C) does.

Let V (∗C) be a nonstandard enlargement of a superstructure V (C). An element x ∈ V (∗C) is called standard if x = ∗X for some X ∈ V (C); internal if x ∈ ∗X for some X ∈ V (C); external if x is not internal.

The following overflow principle is frequently used in the practice of nonstandard analysis. It is based on the fact that the set of standard natural numbers N is not an internal subset of the internal set ∗N of hypernatural numbers. Proposition 3.1 (Overflow Principle). If an internal set A ⊂ ∗N contains arbitrarily large natural numbers then it contains also some infinite hypernatural number.

∗ Deﬁnition 3.2. An element x of C is said to be bounded if for some m0 ∈ N,

|x| ≤ m0; and x is said to be infinitesimal if for every n ∈ N (n 6= 0), |x| ≤ 1/n. Definition 3.3. The set of bounded hypercomplex (resp. hyperreal) numbers is denoted by bC (resp. bR). The set of infinitesimals is denoted by iC (resp. iR). It is well known that bC is a valuation domain and the factor ring bC/iC is isomorphic to C via the standard part mapping. Also, bR is a subring of ∗R which is totally ordered Archimedean valuation domain with maximal ideal iR, which is also an order ideal of bR. The factor ring bR/iR is order isomorphic to R. 3.2. Convex Subrings of ∗C. First, we recall the definition and some properties of convex subrings of ∗C. Definition 3.4. Let F be a nonempty subset of ∗C. We say that F is a convex in ∗C if ∗ (∀x ∈ C)(∀ξ ∈ F)(|x| ≤ |ξ| ⇒ x ∈ F).

Using the fact that any subring of ∗C contains Z, it is clear that if F is a convex subring of ∗C, then F contains bC. We prove that the converse remains true. Proposition 3.5. If M is a bC-submodule of ∗C, then M is convex in ∗C. Proof. Let x ∈ ∗C and ξ ∈ M \{0} such that |x| ≤ |ξ|. Thus x/ξ ∈ bC, and we deduce b ∗ that x = (x/ξ).ξ ∈ C.M ⊆ M, that is, M is convex in C. Corollary 3.6. Let F be a subring of ∗C. Then F is convex if and only if F contains bC. Moreover, if F is convex than any ideal of F is convex. Therefore, any convex subring F of ∗C is a valuation ring.

For the remainder of this paper we ﬁx the following notations : iF denotes the maximal ideal of F, and aF = F \ iF, the set of appreciable elements of F.

Let us recall some properties of iF. Proposition 3.7. Let F be a convex subring of ∗C. Then the following properties hold. (1) iF consists of infinitesimals only (i.e., iF ⊆ iC). (2) iF is a convex ideal in F (i.e., if x ∈ F and ξ ∈ iF, (|x| ≤ |ξ| ⇒ x ∈ iF)). (3) F is a field if and only if F = ∗C. Now, we give some examples of convex subrings of ∗C. 3.2.1. Examples. [24] (i) (Bounded numbers). The ring of bounded nonstandard complex numbers bC is a convex subring of ∗C. Its maximal ideal is iC, the set of infinitesimals. (ii) (Nonstandard complex numbers). The field of the complex numbers ∗C is (trivially) a convex subring of ∗C. Its maximal ideal is {0}. (iii) (Robinson rings). Let ρ be a positive infinitesimal in ∗R. The ring of the ρ-moderate nonstandard numbers is defined by ∗ −n Mρ = {x ∈ C : |x| ≤ ρ for some n ∈ N}. ∗ Then Mρ is a convex subring of C, its maximal ideal is given by i ∗ n Nρ := Mρ = {x ∈ C : |x| ≤ ρ for all n ∈ N}. (iv) (Logarithmic rings) Let ρ be a positive infinitesimal in ∗R. We define ∗ −1/n Lρ = {x ∈ C : |x| ≤ ρ for all n ∈ N≥1}, ∗ then Lρ is a convex subring of C with maximal ideal i ∗ 1/n Lρ = {x ∈ C : |x| ≤ ρ for some n ∈ N≥1}. ∗ (v) (Logarithmic-exponential rings) Let ρ be a positive infinitesimal in R and let Eρ be the smallest convex subring of ∗C containing all iterated exponentials of ρ−1, that is, ∗ −1 Eρ = {x ∈ C : |x| ≤ expn(ρ ) for some n ∈ N},

∗ where exp0(x) = x and expn(x) = exp(expn−1(x)) for x ∈ C and n > 0. The maximal ideal of Eρ is

i ∗ 1 Eρ = {x ∈ C : |x| ≤ −1 , for all n ∈ N}. expn(ρ ) We call Lρ logarithmic rings because | ln ρ| is a typical element of Lρ. The numbers 1/ρ e , | ln ρ| are both in Eρ hence, Eρ are called logarithmic-exponential rings. Deﬁnition 3.8. A convex subring of ∗C is called an exponential ring, if exp(F) ⊆ F. Clearly Eρ is an exponential ring whereas Mρ and Lρ are not exponential rings, as 1/ρ e 6∈ Mρ and exp(| ln ρ|) = 1/ρ 6∈ Lρ.

As bC is a valuation ring, we recall that there is an anti-correspondence between the prime spectrum of bC and the set of convex subrings of ∗C. Indeed, let p be a prime ideal b b b ∗ of C, then ( C)p, the localization of C at p, is a convex subring of C, moreover, its b ∗ maximal ideal is given by p( C)p = p. Conversely, if F is a convex subring of C, then its i b b maximal ideal F is a prime ideal of C and ( C)iF = F.

The following proposition shows that there is an elementary algebraic description of the examples (iii) and (iv) in 3.2.1. The property (iii) follows immediately from Theorem 2.9. Proposition 3.9. Let ρ be a positive inﬁnitesimal, then b 2 (i) Mρ is the localization of C with respect the multiplicative set {1, ρ, ρ ,..., }, that [ 1 \ is, = b and its maximal ideal is given by N = ρn b . Mρ ρn C ρ C n≥0 n≥1 i p b b (ii) Lρ = ρ C is the radical of the ideal generated by ρ in C. i i (iii) Nρ ⊂ Lρ, moreover Lρ is the immediate successor of Nρ in the prime spectrum of bC. i b Thus Lρ is the smallest prime ideal of C containing ρ and Nρ is the largest prime ideal of bC which does not contain ρ.

Using convex subrings of ∗C, a variety of fields Fb is constructed by Todorov [24]. These fields are called F-asymptotic hulls and their elements F-asymptotic numbers. This con- struction can be viewed as a generalization of A. Robinson’s theory of asymptotic numbers, see Lightstone-Robinson [18]. Definition 3.10. Let F be a convex subring of ∗C. The F-asymptotic hull is the factor ring Fb = F/iF. Let stb : F −→ Fb stand for the corresponding quotient mapping, called the quasi-standard mapping.

If x ∈ F, we shall often write xb instead of st(b x) for the quasi-standard part of x.

If F is a convex subring of ∗R, we can deﬁne an order relation in Fb, inherited from the order in ∗R, by

xb ≤ yb if there are representatives x, y with x ≤ y.

Using the convexity of F, the following proposition is straightforward Proposition 3.11. If F is a convex subring of ∗R, then (Fb, ≤) is a linearly ordered field. We note that Todorov [24] proved a strong form of Proposition 3.11 claiming that Fb is a real closed field, that is, Fb is elementarily equivalent to the real numbers. In other words, it has the same first-order properties as the reals: any sentence in the first-order language of fields is true in Fb if and only if it is true in the reals.

We remark that there is a one-to-one correspondence between convex subrings of ∗C and those of ∗R: let F be a convex subring of ∗C, then F0 = F ∩ ∗R is a convex subring of ∗R. Conversely, let F0 be a convex subring of ∗R, then F = {a ∈ ∗C : |a| ∈ F0} is a convex subring of ∗C. Hence F 7→ F ∩ ∗R is one-to-one order preserving correspondence between convex subrings of ∗C and those of ∗R. Moreover, if F is a convex subring of ∗C, then ∗ ∗ F = (F ∩ R) + i (F ∩ R).

3.2.2. Valuations on Fb. Deﬁnition 3.12. We say that a valuation v on Fb is compatible with the absolute value if it satisﬁes ∀ x, y ∈ Fb : |x| ≤ |y| =⇒ v(x) ≥ v(y). Proposition 3.13. A valuation v on Fb is compatible with the absolute value if and only if b C ⊆ {x ∈ F : v(xb) ≥ 0}. b Proof. =⇒ : Let x ∈ C, then there exists n ∈ N such that |x| ≤ n, hence |xb| ≤ n. Since the valuation v is compatible with the absolute value, we obtain v(xb) ≥ v(n) ≥ 0. a ⇐= : Let x, y ∈ Fb such that 0 < |x| ≤ |y|, then there exists x1 and y1 in F, representatives of x and y respectively such that |x1| ≤ |y1|. This shows that x1/y1 is bounded. Thus v(x/y) ≥ 0, that is, v(x) ≥ v(y). b a From the inclusion C ⊆ {x ∈ F : v(xb) ≥ 0}, we obtain C ⊆ {x ∈ F : v(xb) = 0}, where aC = bC \ iC, the set of appreciable numbers. Proposition 3.14. If Fb has a nontrivial real-valued valuation v, then (i) {x ∈ F : v(xb) ≥ 0} is a maximal subring of F. (ii) Moreover, if v is compatible with the absolute value, then [ 1 = = b , F Mα αn C n∈N

a where α is any element in F+, such that 0 < v(αb) < +∞. Proof. (i) Let G be a subring of F such that G ⊃ {x ∈ F : v(xb) ≥ 0}. Then there exists i b ∈ G such that v(b) < 0. Since F = {x ∈ F : v(xb) = +∞} ⊆ G, we have only to show a a xb x that ⊆ . Let x ∈ , then v( ) = v(x) − nv(b) > 0 for n large. Hence n ∈ , that F G F bn b b G is, x ∈ bnG. Thus F ⊆ G. a i i a (ii) Let α ∈ F+ such that 0 < v(αb) < +∞. Hence α ∈ R+ \ F. Let x ∈ F, then there exists n ∈ N such that v(1/αn) < v(x). By compatibility of the valuation v with the 1 b b 1 absolute value, we get |x| < n . Thus |x| < n . b αb α ρ Example 3.15. The Robinson valuation v(x) = st(logρ(|x|)) on C := Mρ/Nρ is compatible with the absolute value. The following result provides a description of the prime spectrum of bC. Theorem 3.16. ( ) b \ i i (i) Spec( C) = Nρ : T ⊂ R+ \{0} ∪ { C} . ρ∈T (ii) Let p be a nonzero and a non maximal prime ideal of bC. Then the following are equivalent (a) p has an immediate successor in the prime spectrum of bC. i (b) p = Nρ for some ρ ∈ R+ \{0}. b (c) The residue field ( C)p/p has a nontrivial real valuation compatible with the absolute value. (iii) The zero ideal has no immediate successor, and the maximal ideal iC has no immediate predecessor. b i (iv) Consecutive prime ideals of C are of the form Nρ ⊂ Lρ. i (v) Nρ has no immediate predecessor and Lρ has no immediate successor; in particular there are no three consecutive elements in Spec(bC). Before giving the proof, we show the following lemmas. Lemma 3.17. b i [ (1) The maximal ideal of C satisfies the equality C = Nρ, so it has no im- i ρ∈ R+\{0} mediate predecessor. i (2) For all ρ ∈ R+ \{0}, Nρ 6= (0) (so (0) has no immediate successor). \ (3) (0) = Nρ. i ρ∈ R+\{0} Proof. Clearly, (2) and (3) hold. i S i We have only to show that ⊂ i N . Let x ∈ and consider C ρ∈ R+\{0} ρ C ∗ 1/n A = {n ∈ N : |x| < 1/n}. A is an internal subset of ∗R containing N, hence A contains some infinite integer N ∈ N∞. 1/N i n Let ρ := |x| . Since ρ < 1/N, we get ρ ∈ R and |x| ≤ ρ for all n ∈ N, i.e., x ∈ Nρ.

Lemma 3.18. Let ρ be a positive infinitesimal. Then there is no positive infinitesimal τ i such that Lρ = Nτ . Proof. By contradiction, assume that there exists τ ∈ iR \{0} such that p b ρ C = Nτ . n In particular, we obtain ρ ∈ Nτ , that is ρ ≤ τ for all n ∈ N. Hence, by the overflow principle, there exists N infinite such that ρ ≤ τ N . Moreover for every infinite positive M n M p b real number M, we have τ ≤ τ for all n ∈ N, that is, τ ∈ Nτ = ρ C. Hence there exist n ∈ and x, a positive bounded number, such that τ nM = ρx. N √ √ √ In particular, for M = N, we have τ n N = ρx ≤ ρ. Hence √ τ 2n N ≤ ρ. Combining with the estimates ρ ≤ τ M , we obtain √ τ 2n N ≤ τ N . √ This implies that 2n N ≥ N, which contradicts the fact that N is infinite.

Proposition 3.19. Let ρ and τ be two positive inﬁnitesimal numbers. If Nρ ⊂ Nτ , then p b there exist inﬁnitely many prime ideals Nx such that Nρ ⊂ ρ C ⊂ Nx ⊂ Nτ .

p b Proof. As ρ C is the immediate succesor of Nρ and using Lemma 3.18, we obtain

p b Nρ ⊂ ρ C ⊂ Nτ . p b Let x1 ∈ Nτ \ ρ C, we have

p b ρ C ⊂ Nx1 ⊂ Nτ . Indeed, the ﬁrst inclusion is a consequence of Lemma 2.4. For the second inclusion, we b have Nx1 ⊂ x1 C ⊂ Nτ .

Applying the same process to the prime ideals Nx1 ⊂ Nτ , we construct an inﬁnite

sequence of prime ideals Nxi satisfying

p b p b p b Nρ ⊂ ρ C ⊂ Nxi ⊂ xi C ⊂ Nxi+1 ⊂ xi+1 C ⊂ Nτ Proof of Theorem 3.16. (i) True for any divided domain (see Theorem 2.3). (ii) We have only to show that (b) is equivalent to (c). If p = Nρ, then the residue ﬁeld b ρ ( C)p/p is C, the Robinson ﬁeld, which has a valuation compatible with the absolute value, see Example 3.15. The converse is a consequence of Proposition 3.14 (ii). (iii) By Lemma 3.17. (iv) Follows from Theorem 2.9. (v) This follows immediately from Lemma 3.18.

Remark 3.20. The spectrum of bC has three kinds of prime ideals ↓ i • Lρ | ↓ •Nρ •p ↑ ↑ Consecutive prime ideals Prime ideals with neither immediate successor nor immediate predecessor As a class of examples of prime ideals of bC with neither immediate successor nor immediate predecessor, one can mention the ideals

i ∗ 1 Eρ = {x ∈ C : |x| ≤ −1 , for all n ∈ N}, expn(ρ )

where ρ is a positive inﬁnitesimal number. Indeed, Eρ is an exponential ring whereas Mρ and Lρ are not exponential. Moreover, we have the following containments;

i i p b Eρ ⊂ Nρ ⊂ Lρ = ρ C. Corollary 3.21. Let F be a convex subring of ∗C. Then the following properties hold (i) Every nonzero prime ideal of F is of inﬁnite height. (ii) Let p be a non maximal prime ideal of F. Then coht(p) = 1 if and only if F = Lρ and p = Nρ, otherwise coht(p) = +∞.

We close the paper by a topological property of Spec(bC).

Recall that a topological space is said to be solvable [6], if it is the disjoint union of two dense subsets. Clearly Spec(bC) is solvable since {(0)} is dense and Spec(bC) \{(0)} is also dense as it contains Gold(bC), see Remark 2.10 and Lemma 3.17 (2). The following theorem provides a ﬁner decomposition of Spec(bC). Theorem 3.22. The space Spec(bC) (endowed with the Zariski topology) is the disjoint union of zero and three dense subsets.

Proof. As {(0)} is dense in Spec(bC), we will show that non-zero elements of Spec(bC) are the disjoint union of three subsets X,Y,Z, where X is strongly dense and Y,Z are dense in Spec(bC). i We let X = {Nρ : ρ is a positive inﬁnitesimal number} ∪ { C}, p Y = { ρ bC : ρ is a positive inﬁnitesimal number}, and Z = Spec(bC) \ (X ∪ Y ∪ {0}); this means that Z is the set of all nonzero prime ideals of Spec(bC) having neither an immediate successor nor an immediate predecessor. As X = Gold(bC), X is strongly dense in Spec(bC), see Remark 2.10.

Now, let us show that Y,Z are dense. Let O be a nonempty open set of Spec(bC). As (0) has no immediate successor, O 6= {(0)}. Hence consider a nonzero prime ideal p ∈ O, and ρ a positive inﬁnitesimal number in p \{0}. Then

i i p b Eρ ⊂ Nρ ⊂ Lρ = ρ C ⊆ p. i i As O is Alexandroﬀ-open, we get Eρ ∈ (Z ∩ O) and Lρ ∈ (Y ∩ O), showing that Y and b Z are dense subsets of Spec( C). It would be of great interest to further investigate and characterize spectral spaces or topological spaces which are the disjoint union of strongly dense and dense subsets.

Acknowledgement The authors would like to thank the anonymous reviewers for their helpful comments that greatly contributed to improving the ﬁnal version of the paper.

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Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia E-mail address: [email protected] E-mail address: [email protected]