Hilbert Ring Research Articles

On maximal ideals of the polynomial ring and some conjectures on Ext-index of rings

Recall that a ring [Formula: see text] is a Hilbert ring if any maximal ideal of [Formula: see text] contracts to a maximal ideal of [Formula: see text]. The main purpose of this paper is to characterize the prime ideals of a commutative ring [Formula: see text] which are traces of the maximal ideals of the polynomial ring [Formula: see text]. In this context, we prove that if [Formula: see text] is a prime ideal of [Formula: see text] such that [Formula: see text] is a semi-local domain of (Krull) dimension [Formula: see text], then [Formula: see text] is the trace of a maximal ideal of [Formula: see text]. Whereas, if [Formula: see text] is Noetherian and either ([Formula: see text]) or (the quotient field of [Formula: see text] is algebraically closed, [Formula: see text] and [Formula: see text] is not semi-local), then [Formula: see text] is never the trace of a maximal ideal of [Formula: see text]. Putting these results into use in investigating the Ext-index of Noetherian rings, we establish connections between the finiteness of the Ext-index of localizations of the polynomial rings [Formula: see text] and the finiteness of the Ext-index of localizations of [Formula: see text]. This allows us to provide a new class of rings satisfying some known conjectures on Ext-index of Noetherian rings as well as to build bridges between these conjectures.

Hilbert rings with maximal ideals of different heights and unruly Hilbert rings

AbstractLet $f:R\to S$ be a ring homomorphism and J be an ideal of S. Then the subring $R\bowtie ^fJ:=\{(r,f(r)+j)\mid r\in R$ and $j\in J\}$ of $R\times S$ is called the amalgamation of R with S along J with respect to f. In this paper, we characterize when $R\bowtie ^fJ$ is a Hilbert ring. As an application, we provide an example of Hilbert ring with maximal ideals of different heights. We also construct non-Noetherian Hilbert rings whose maximal ideals are all finitely generated (unruly Hilbert rings).

A Zariski topology on integrally closed maximal subrings of a commutative ring

Let R be a commutative ring and Xi.c (R) denotes the set of all integrally closed maximal subrings of R. It is shown that if R is a non-field G-domain, then there exists S ∈ Xi.c (R) with (S : R) = 0. If K is an algebraically closed field which is not absolutely algebraic, then we prove that the polynomial ring K[X] has an integrally closed maximal subring with zero conductor too; a characterization of integrally closed maximal subrings of K[X] with (non-)zero conductor is given. It is observed that, an integrally closed maximal subrings S of K[X] is a principal ideal domain (PID) if and only if M = Sq for some q− 1 ∈ K \\ S, where M is the crucial maximal idea of the extension S ⊆ K[X]. We show that if f (X, Y) is an irreducible polynomial in K[X, Y ], then there exists an integrally closed maximal subring S of K[X, Y ] with (S : K[X, Y ]) = f (X, Y) K[X, Y ]. It is proved that, if R is a ring and , where I is an ideal of R, then is an ideal of R} is a topology for closed sets on X (R). We show that this space has similar properties such as those one in the Zariski spaces on Spec(R) or Kn (the affine space). In particular, if K is a field which is not algebraic over its prime subring, then Xi.c (K[X 1,…, Xn ]) is irreducible and if in addition K is algebraically closed, then we prove a similar full form of the Hilbert Nullstellensatz for K[X 1,…, Xn ]. Moreover, if R is a non-field G-domain or R = K[X], where K is an algebraically closed field which is not algebraic over its prime subring, then ∅ ≠ gen(X i.c (R)) = {S ∈ Xi.c (R) | (S : R) = 0}. We determine exactly when the space Xi.c (R) is a Ti-space for i = 0, 1, 2. In particular, we show that if Xi.c is T1-space then R is a Hilbert ring and |Xi.c (R)| ≤ 2|Max(R)|. Finally, we determine when the space Xi.c is connected.

Hilbert rings and G(oldman)-rings issued from amalgamated algebras

Let [Formula: see text] and [Formula: see text] be two rings, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] be a ring homomorphism. The ring [Formula: see text] is called the amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text]. It was proposed by D’anna and Fontana [Amalgamated algebras along an ideal, Commutative Algebra and Applications (W. de Gruyter Publisher, Berlin, 2009), pp. 155–172], as an extension for the Nagata’s idealization, which was originally introduced in [Nagata, Local Rings (Interscience, New York, 1962)]. In this paper, we establish necessary and sufficient conditions under which [Formula: see text], and some related constructions, is either a Hilbert ring, a [Formula: see text]-domain or a [Formula: see text]-ring in the sense of Adams [Rings with a finitely generated total quotient ring, Canad. Math. Bull. 17(1) (1974)]. By the way, we investigate the transfer of the [Formula: see text]-property among pairs of domains sharing an ideal. Our results provide original illustrating examples.

Fractal Minkowski Antenna Loaded with Hilbert Curve as Complementary Split Ring Resonator

Fractal Minkowski patch antenna is proposed. Classical complementary split ring resonator (CSRR) is slotted on the patch. For comparison, fractal Hilbert shape split ring resonator also slotted on the patch. These slots make the patch behaves as a left-handed material in certain frequency band. Minkowski fractal antenna has three resonance frequencies. The complementary split ring resonator and Hilbert ring affects the upper two frequencies and make the antenna resonate at lower frequencies. Lowering the resonance frequencies cause a reduction in antenna size. More reduction of antenna size is obtained when fractal Hilbert is used as complementary split ring resonator. The simulated results have been done by using (HFSS) software, which is based on finite element modeling. The measurement of fabricated antenna shows good agreement with simulation results

Fractal Minkowski Antenna Loaded with Hilbert Curve as Complementary Split Ring Resonator

Fractal Minkowski patch antenna is proposed. Classical complementary split ring resonator (CSRR) is slotted on the patch. For comparison, fractal Hilbert shape split ring resonator also slotted on the patch. These slots make the patch behaves as a left-handed material in certain frequency band. Minkowski fractal antenna has three resonance frequencies. The complementary split ring resonator and Hilbert ring affects the upper two frequencies and make the antenna resonate at lower frequencies. Lowering the resonance frequencies cause a reduction in antenna size. More reduction of antenna size is obtained when fractal Hilbert is used as complementary split ring resonator. The simulated results have been done by using (HFSS) software, which is based on finite element modeling. The measurement of fabricated antenna shows good agreement with simulation results

Modules satisfying the prime and maximal radical conditions

In this paper, we introduce and study $\mathbb P$-radical and $\mathbb{M}$-radical modules over commutative rings. We say that an $R$-module $M$ is \textit{$\mathbb P$-radical} whenever $M$ satisfies the equality $(\sqrt[p]{{\mathcal{P}}M}:M)=\sqrt{{\mathcal{P}}}$ for every prime ideal ${\mathcal{P}}\supseteq Ann({\mathcal{P}}M)$, where $\sqrt[p]{{\mathcal{P}}M}$ is the intersection of all prime submodules of $M$ containing ${\mathcal{P}}M$. Among other results, we show that the class of $\mathbb P$-radical modules is wider than the class of primeful modules (introduced by Lu \cite{Lu4}). Also, we prove that any projective module over a Noetherian ring is $\mathbb P$-radical. This also holds for any arbitrary module over an Artinian ring. Furthermore, we call an $R$-module $M$ by $\mathbb{M}$-\textit{radical} if $(\sqrt[p]{{\mathcal{M}}M}:M)={\mathcal{M}}$, for every maximal ideal ${\mathcal{M}}$ containing ${\rm Ann\,}(M)$. We show that the conditions $\mathbb P$-radical and $\mathbb{M}$-radical are equivalent for all $R$-modules if and only if $R$ is a Hilbert ring. Also, two conditions primeful and $\mathbb{M}$-radical are equivalent for all $R$-modules if and only if $\mbox{dim\,}(R)=0$. Finally, we remark that the results of this paper will be applied in a subsequent work of the authors to construct a structure sheaf on the spectrum of $\mathbb P$-radical modules in the point of algebraic geometry view.

MODULES WHOSE CLASSICAL PRIME SUBMODULES ARE INTERSECTIONS OF MAXIMAL SUBMODULES

Commutative rings in which every prime ideal is an intersection of maximal ideals are called Hilbert (or Jacobson) rings. We propose to define classical Hilbert modules by the property that classical prime submodules are intersections of maximal submodules. It is shown that all co-semisimple modules as well as all Artinian modules are classical Hilbert modules. Also, every module over a zero-dimensional ring is classical Hilbert. Results illustrating connections amongst the notions of classical Hilbert module and Hilbert ring are also provided. Rings R over which all modules are classical Hilbert are characterized. Furthermore, we determine the Noetherian rings R for which all finitely generated R-modules are classical Hilbert.

NONCOMMUTATIVE HILBERT RINGS

Commutative rings in which every prime ideal is the intersection of maximal ideals are called Hilbert (or Jacobson) rings. This notion was extended to noncommutative rings in two different ways by the requirement that prime ideals are the intersection of maximal or of maximal left ideals, respectively. Here we propose to define noncommutative Hilbert rings by the property that strongly prime ideals are the intersection of maximal ideals. Unlike for the other definitions, these rings can be characterized by a contraction property: R is a Hilbert ring if and only if for all n∈ℕ every maximal ideal [Formula: see text] contracts to a maximal ideal of R. This definition is also equivalent to [Formula: see text] being finitely generated as an [Formula: see text]-module, i.e., a liberal extension. This gives a natural form of a noncommutative Hilbert's Nullstellensatz. The class of Hilbert rings is closed under finite polynomial extensions and under integral extensions.

Strongly real hilbert rings and the real jacobson semisimplicity

In this paper, several false results in reference [1] related to rela Hilbert rings and the ral Jacobson semisimplicity are negated by a counterexample. By introducing the notion of a strongly real Hilbert ring, we characterize those rings of which every finitely generated real extension is real jacobson semisimple. Moreover, the so-called strictly real Hilbert rings are considered.

The Prime Avoidance of Ideals in Noetherian Hilbert Rings

It is shown that the polynomial ring R[X] over a Noetherian Hilbert ring R is coprimely packed if and only if R is Artinian.

The Jacobson radical of commutative semigroup rings

In this paper we consider semiprimitive commutative semigroup rings and related matters. A ring is said to be semiprhnitive if the Jacobson radical of it is equal to zero. This property is one of the most important in the theory of semigroup rings, and there is a prolific literature pertaining to the field (see 1,,14]). All semiprimitive rings are contained in another interesting class of rings. Let 8 denote the class of rings R such that ~/(R) = B(R), where J and B are the Jacobson and Baer radicals. Clearly, every semiprimitive ring is in 6. This class, appears, for example, in the theory of Pl-rings and in commutative algebra. (In particular, every finitely generated PI-ring and every Hilbert ring are in 6.) Therefore, it is of an independent interest. Meanwhile it is all the more interesting because any characterization of the semigroup rings in 6 will immediately give us a description of semiprimitive semigroup rings. Indeed, a ring R is semiprimitive if and only if R~6 and R is semiprime, i.e., B(R)=O. Semiprime commutative semigroup rings have been described by Parker and Giimer 1,12] and, in other terms, by Munn [9]. So it suffices to characterize semigroup rings in 6. Semigroup rings of 6 were considered by Karpilovsky r5], Munn 1,6-9], Okninski 1-10-h and others. In this paper commutative semigroup rings which are in 6 will be described completely. To this end one should know the structure of the Jacobson radical J(R[S]). In [2] Jespers described J(R1,S]) under rather weak assumptions on R. They hold, in particular, for every commutative R. Here we shall give another (quite short) description of J(R1,S]) which does not require any restriction on R. Besides, it is specially fitted for testing whether an element is in J(R1,S]), and this is essential for our proofs.

Induced resolutions and grothendieck groups of polycyclic-by-finite-groups

Let Г be a polycyclic-by-finite group, R a commutative Noetherian ring, G 0( RГ) the Grothendieck group of finitely generated RГ-modules, and G 0( RГ, F) the subgroup generated by the classes of modules induced from finite subgroups of Г. It is expected that G 0 (RГ)=G 0 (RГ, F ) . As partial evidence for this, we show that G 0 (RГ) G 0 (RГ, F ) is torsion, with an explicit bound for the exponent, in the case where Г is abelian-by-finite and R a regular Noetherian Hilbert ring of finite global dimension.

The rings R( X) and R〈 X〉

The rings R( X) and R〈 X〉 are investigated. Of particular importance is the question of when R( X) or R〈 X〉 is an arithmetical ring, a Prüfer ring, a Hilbert ring or a Euclidean ring. Answers are sought in terms of the base ring R.

Commutative Monoid Rings as Hilbert Rings

Let $S$ be a cancellative monoid with quotient group of torsion-free rank $\alpha$. We show that the monoid ring $R(S)$ is a Hilbert ring if and only if the polynomial ring $R[{\{ {X_i}\} _{i \in I}}]$ is a Hilbert ring, where $\left | I \right | = \alpha$.

Polynomial rings over a Hilbert ring.

Polynomial rings over a Hilbert ring.

Modules over polycyclic groups have many irreducible images

Recall that a Noetherian ring R is a Hilbert ring if the Jacobson radical of every factor ring of R is nilpotent. As one of the main results of [13], J. E. Roseblade proved that if J is a commutative Hilbert ring and G is a polycyclic-by-finite group then JG is a Hilbert ring. The main theorem of this paper is a generalisation of this result in the case where all the field images of J are absolute fields—we shall say that J is absolutely Hilbert. The result is stated in terms of the (Gabriel–Rentschler–) Krull dimension; the definition and basic properties of this may be found in [5]. Let M be a finitely generated right module over the ring R. We write AnnR(M) (or just Ann(M)) for the ideal {r ∈ R: Mr = 0}, the annihilator of M in R. If M is also a left module, its left annihilator will be denoted l-AnnR(M). If R is a group ring JG, put

R Noetherian implies R〈 X〉 is a Hilbert Ring

R Noetherian implies R〈 X〉 is a Hilbert Ring

Hilbert subalgebras of finitely generated algebras, II

Recall that a Hilbert ring (also called a Jacobson ring) is a commutative ring in which every prime ideal is an intersection of maximal ideals. Such rings were considered by Goldman [4] and Krull [6] to obtain an abstract formulation of Hilbert’s Nullstellensatz. They each proved the basic theorem that any finitely generated algebra over a Hilbert ring is again a Hilbert ring. Krull also proved that every co2mtabZy generated algebra over an uncountable field is a Hilbert ring. (This was subsequently generalized by Lang [7] and Gilmer [3].) We consider here an integral domain R which is a &algebra of a finitely generated algebra over a field F, and ask whether R must be a Hilbert ring. We prove that if P is a prime ideal of R, with t.d. R/P + rank P = t.d. R, then R/P is contained in a finitely generated F-algebra (t.d. = transcendence degree over F). Hence P is an intersection of maximals. It follows that if t.d. R 3 and F is countable, then R need not be Hilbert and, indeed, we give a construction showing that R contains a non-Hilbert subalgebra. When F is uncountable, Krull’s theorem shows that R is Hilbert, regardless of its transcendence degree. (For, R has countable dimension as an F-vector space.) We give analogous results for algebras over a discrete valuation ring, and, in the Appendix, establish when all subalgebras of a finitely generated algebra are finitely generated. The author would like to thank his colleague Lance Small for a number of constructive conversations, and George Bergman for some illuminating comments on the examples. In addition he thanks the referee for pointing out the application of Nagata’s work on ideal transforms to the proof of Theorem 1.

Hilbert integral domains with maximal ideals of preassigned height

A commutative ring with identity R is called a Hilbert ring if every prime ideal in R is an intersection of maximal ideals. It is well known that, if R is a Hilbert ring, then the polynomial ring R[X] is also a Hilbert ring. Indeed, Hilbert rings first arose in this connection: the fact that R is Hilbert implies R[X] is Hilbert, being used by Krull in [4] and Goldman in [l] to give a ring theoretic proof for the Hilbert Nullstellensatz. If R is a Hilbert ring, then every maximal ideal in R[X] lies over a maximal ideal in R (see, for example, [3, p. 181). Thus, if R is a Hilbert ring and every maximal ideal in R has height >n, then every maximal ideal in R[XJ has height >n + 1. (A prime ideal P in R is said to have height n if there is a chain P,, < P1 < -*a < P, = P of prime ideals in R but no such chain of longer length.) The most naturally occurring examples of Hilbert domains are finitely generated ring extensions of a field or the ring of integers. Such examples as these have the property that all maximal ideals have the same height. Leslie Roberts in [5] points out that Hilbert domains with maximal ideals of different height do not seem to be well known, and gives, by means of an interesting localization technique, examples of Hilbert domains having maximal ideals of height both n and n + 1. The purpose of this note is to construct Hilbert domains with maximal ideals of various preassigned height. This is done by making use of intersection properties for certain integral domains given by Jack Ohm and the author in [2].