© Hindawi Publishing Corp. NAGATA RINGS AND DIRECTED UNIONS OF ARTINIAN SUBRINGS

We investigate when a Nagata ring R(X) can be written as a directed union of Artinian subrings. For a family of zero-dimensional rings {Rα}α∈A, we show that ∏α∈ARα(Xα) is not a directed sum of Artinian subrings.


Introduction.
All rings considered in this paper are assumed to be commutative with a unit element.If R is a subring of a ring S, we assume that the unity element of S belongs to R, and hence is the unit element of R. Let Spec(R), Z(R), Inv(R), and Ann R (I) denote, respectively, the spectrum of R (the set of prime ideals of R), the set of zero-divisors of R, the set of invertible elements of R, and the annihilator of a subset I of R. By the dimension of R, denoted as dim R, we mean the Krull dimension: dim R is the maximal length of a chain of proper prime ideal P 0 ⊂ P 1 ⊂ ••• ⊂ P n of R. If there is no upper bound on the length of such chains, then we write dim R = ∞.
In this paper, we study zero-dimensional rings, in which each proper prime ideal is maximal, and Nagata rings.Our attention will be focused on proving that an infinite direct product of R α (X α ), where {R α } α∈A is a family of zerodimensional rings and {X α } α∈A is a family of indeterminates, is not a directed union of Artinian subrings.Rings of Krull dimension zero have been studied intensively in the literature since the sixties.Directed unions of Artinian subrings have been investigated more recently, see [5,7].
In [1, Problem 42], Gilmer and Heinzer raised the following question.(Q) Under which conditions is a von Neumann regular ring a directed union of Artinian subrings?It would be interesting to consider this question for Nagata rings.
In 1992, Gilmer and Heinzer showed that a product of zero-dimensional rings has dimension zero or infinity, see [6,Theorem 11].
Let R be a commutative ring and f ∈ R[X].The content of f is the ideal σ (f ) of R generated by the coefficients of f .Then is a multiplicatively closed subset of R[X], and the localization R(X) = S −1 R[X] is called the Nagata ring in one variable over R. The Nagata ring in n variables with coefficients in R is the ring where 2. Nagata rings.We first fix notation.Data will consist of a directed system (R j ,f jk ) of rings indexed by a directed set (I, ≤) and its directed union R = j∈I R j , together with the canonical maps f j : R j → R. The ring R is a directed union of R j 's corresponding to the f jk 's being inclusion maps.Thus directed unions can be treated by assuming all f jk to be monomorphisms.Notice that if R j is a ring for each j ∈ I, then R is also a ring.However, R is not necessarily Noetherian even if each R j is Noetherian.If R = j∈I R j is a directed union of Artinian subrings, then we regard each R i as a subring of R; in particular, R i and R have the same unit element.
The proof of Lemma 2.1 is straightforward and is left to the reader.
Lemma 2.1.Let {R i } k i=1 be a finite family of rings and X a variable.Then Let K and L be two fields.From Lemma 2.1, we know that and we can view a polynomial in , and hence we have the following result.
is an infinite family of rings and X a variable over α∈A R α , then ϕ : Before proving Proposition 2.3, we need the following lemma.
, is an injective homomorphism.Now, we consider The converse of Proposition 2.3 is not true in general, as shown in the next example.
Example 2.5.For each i ∈ Z + , let R i = F p i be the Galois field with p i elements, where p is a positive prime integer.From [10, Theorem 5.5, page 247], F p i = GF(p)(ξ i ), where ξ i is a p i th primitive root of unity, for each i ∈ Z + , and GF(p) is the Galois field with p elements.Let Our next result will be useful later.Lemma 2.6.Let R be a zero-dimensional ring with finite spectrum.Then R can be expressed as a finite product of zero-dimensional quasilocal subrings.
(2) If R(X) is a directed union of Artinian subrings, then, by [8, Theorem 2.4(a)], each R j = S j ∩R is zero dimensional.Since R j ⊆ S j and Spec(S j ) is finite (cf.[3,Theorem 8.3]), this yields that each Spec(R j ) is finite.As R is reduced, and by [4, Theorem 3.1], each R j is a von Neumann regular ring with finite spectrum.It follows that R j is Artinian, and hence, R = j∈I R j is a directed union of Artinian subrings.
(3) The proof of this result follows from the fact that Spec Remark 2.8.(1) Let R be a hereditary zero-dimensional ring, that is, a ring for which all subrings are zero dimensional.Then R is a directed union of Artinian subrings, and therefore, R(X) is a directed union of Artinian subrings that is not hereditarily zero dimensional, since R[X] ⊂ R(X) and dim(R[X]) = 1 (cf.[13,Theorem 2]).
(2) Let R be a von Neumann regular ring and X 1 ,...,X n variables over R. We denote R(X 1 ,...,X n ) = R(n) for each n ∈ Z + .Then, by Proposition 2.7 and [9, Lemma 15.3], R is a directed union of Artinian subrings if and only if R(n) is a directed union of Artinian subrings, for each n ∈ Z + .
(3) If, in Proposition 2.7(3), we take X = {Y i } i∈I an infinite family of indeterminates over R, then R is a directed union of zero-dimensional subrings with finite spectra if and only if so is R(X).
Let η(x) be the index of nilpotency of x ∈ R. We define where N(R) is the set of nilpotent elements of R. From [7, Theorem 3.4], we know that dim( α∈A T α ) = 0 if and only if {α ∈ A | η(T α ) > k} is finite for some k ∈ Z + , where {T α } α∈A is a family of zero-dimensional rings.
Let R be a ring such that η(R) < k for some k ∈ Z + and let X be a variable over R. Then η(R[X]) need not be bounded.Also, we note that if dim(R) = 0, then dim(R(X)) = 0.
Let {R α } α∈A be a family of zero-dimensional rings and X a variable, and suppose that dim( α∈A R α ) = 0. We added that, if each R α is a directed union of Artinian subrings, then R α (X) is also a directed union of Artinian subrings for each α ∈ A, see Proposition 2.7.Assume that there exists k ∈ Z + such that {α ∈ A | η(R α ) > k or there exists M ∈ Spec(R α ) : |R α /M| > k} is finite.Then, by [7,Theorem 6.7], α∈A R α is a directed union of Artinian subrings.However, for each k ∈ Z + , {α ∈ A | η(R α (X)) > k} is an infinite set.This means that α∈A R α (X) is not zero dimensional.Now, we suppose that each R α is a von Neumann regular ring, and we show that R α (X) is also a von Neumann regular ring, and hence, α∈A R α (X α ) is a von Neumann regular ring, where each X α is a variable over R α .Theorem 2.9.Let {R α } α∈A be a family of von Neumann regular rings and X α an indeterminate over R α , for each α ∈ A. Then α∈A R α (X α ) is not a directed union of Artinian subrings.
The proof of Theorem 2.9 requires the following two lemmas.Lemma 2.10.Let R be a ring and U a multiplicatively closed subset of R. If R is reduced, then U −1 R is also reduced.
Then there exists n 0 ∈ N * such that (r /s) n 0 = 0; this means that there exists u ∈ U such that (r u) n 0 = 0. Since R is reduced, we have r u = 0, and hence, r /s = 0.In other words, The converse implication follows from the fact that every subring of a reduced ring is reduced.
Note that the two equivalent conditions of Lemma 2.11 are also equivalent to Proof of theorem 2.9.By [7, Theorem 6.7], α∈A R α (X α ) is a directed union of Artinian subrings if and only if there exists k ∈ Z + such that {α ∈ A | there exists M ∈ Spec(R α (X α )) with |R α /M| > k} is finite.It was shown in [12, (6.17) Corollary 2.12.Let {R α } α∈A be a family of von Neumann regular rings and X a variable.Then α∈A R α (X) is not a directed union of Artinian subrings.
Let R be a ring and {R α } α∈A an infinite family of nonzero rings such that R is, up to isomorphism, a subring of each R α .We use R * to denote the image of R under the diagonal imbedding, that is, R * = ϕ(R), where ϕ : R α∈A R α is the monomorphism defined by ϕ(x) = {x α } α∈A such that x α = x for each α ∈ A. Proposition 2.13.Let F be an absolutely algebraic field and R = ω 0 F a countable direct product of copies of F .Define has only finitely many distinct coordinates . (2.3) Then is the maximal subring of R which can be expressed as a directed union of Artinian subrings.
Proof.First, we claim that is a directed union of Artinian subrings.For each j ∈ Z + , we define S j as the subring of consisting of all sequences {x i } ∞ i=0 ∈ such that x j = x j+1 = •••.If we denote by π the prime subring of R, then each S j contains π , S 0 is the diagonal imbedding of F in R, and S j F j+1 is an Artinian von Neumann regular ring.Clearly, S j ⊆ S j+1 for each j ∈ Z + .Therefore, = ∞ j=1 S j and is a directed union of Artinian subrings.Now, let T be a subring of ω 0 F with T = j∈J T j a directed union of Artinian subrings and t = {t i } ∞ i=1 ∈ T .There exists j 0 ∈ J such that t ∈ T j 0 and T j 0 is a finite product of fields; hence t ∈ .
Example 2.14.Let p be a prime integer and X a variable over GF(p), where GF(p) is the Galois field with p elements.Let R = (GF(p)(X)) ω 0 be a countable direct product of copies of GF(p)(X).We note that R is a von Neumann regular ring as a direct product of fields.By Theorem 2.9, the ring R is not a directed union of Artinian subrings.Let = {{x i } ∞ i=1 ∈ R : {x i } ∞ i=1 has only finitely many distinct components} and V = (GF(p)(X)) * + I, where I = ∞ i=1 GF(p) is the direct sum ideal of R and (GF(p)(X)) * is the diagonal imbedding of GF(p)(X) in R. The ring is the biggest subring of R which is a directed union of Artinian subrings.We remark that, if V ⊂ , then, by [11,Corollary 4], V is also a directed union of Artinian subrings.Since GF(p) is a finite field, we have ω 0 GF(p) ⊂ and, by [11,Corollary 4], is a directed union of Artinian subrings.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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First
Round of ReviewsMarch 1, 2009 By the Chinese remainder theorem, where R/S M i (0) is quasilocal and zero-dimensional, for i = 1,...,n.We note that if R is a von Neumann regular ring (i.e., R is reduced and zero dimensional), then R is an Artinian if and only if R is a finite product of fields if and only if R is Noetherian.Indeed, if R is von Neumann regular and Artinian, then, by [3, Corollary 8.2], Spec(R) is finite, and hence, R = R 1 ⊕•••⊕R n , where each R i is a quasilocal and zero-dimensional ring, for i = 1,...,n.Since R is a von Neumann regular ring, each R i is a von Neumann regular ring, by [4, Result 3.2].As R i is a quasilocal ring, by [4, Theorem 3.1], R i is a field for i = 1,...,n, and it follows that R is a finite product of fields.Let R be a ring and X a variable over R. Then (1) if R is a directed union of Artinian subrings, then so is R(X); (2) if R is a reduced ring and R(X) is a directed union of Artinian subrings, then R has the same property; (3) R is a directed union of zero-dimensional subrings with finite spectra if and only if R(X) has the same property.