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Pre-Publication Accepted Manuscript 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 This is a preliminary PDF of the author-produced manuscript that has been peer-reviewed and accepted for publication. It has not been copyedited, proofread, or finalized by AMS Production staff. Once the accepted manuscript has been copyedited, proofread, and finalized by AMS Production staff, the article will be published in electronic form as a \Recently Published Article" before being placed in an issue. That electronically published article will become the Version of Record. This preliminary version is available to AMS members prior to publication of the Version of Record, and in limited cases it is also made accessible to everyone one year after the publication date of the Version of Record. 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 spec- trum 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 pro- vides 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 non- archimedean 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 commu- tative 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. 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. THE SPECTRUM OF BOUNDED COMPLEX NUMBERS 3 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 immedi- ate 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 contain- ment. 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 2 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 2 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 2 R, either a j b or b j 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) = fmg [ Iρ : T is a nonempty subset of m n f0g : ρ2T 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. 4 OTHMAN ECHI AND ADEL KHALFALLAH 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: ρ2mnp Proof. As p is a divided prime ideal; then it is comparable to any principal ideal of R. So, n if ρ 2 m n p, we have p ⊆ ρ R, for all n ≥ 1. This leads to p ⊆ Iρ; and consequently \ \ p ⊆ Iρ.
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