Zero-dimensional Rings Research Articles

Height theorems and unmixedness for finitely generated algebras over zero-dimensional rings

Height theorems and unmixedness for finitely generated algebras over zero-dimensional rings

A Gröbner Basis Algorithm for Ideals over Zero-Dimensional Valuation Rings

Zero-dimensional valuation rings are one kind of non-Noetherian rings. This paper investigates properties of zero-dimensional valuation rings and prove that a finitely generated ideal over such a ring has a Grobner basis. The authors present an algorithm for computing a Grobner basis of a finitely generated ideal over it. Furthermore, an interesting example is also provided to explain the algorithm.

Prime avoidance property

Let [Formula: see text] be a commutative ring, we say that [Formula: see text] has prime avoidance property, if [Formula: see text] for an ideal [Formula: see text] of [Formula: see text], then there exists [Formula: see text] such that [Formula: see text]. We exactly determine when [Formula: see text] has prime avoidance property. In particular, if [Formula: see text] has prime avoidance property, then [Formula: see text] is compact. For certain classical rings we show the converse holds (such as Bezout rings, [Formula: see text]-domains, zero-dimensional rings and [Formula: see text]). We give an example of a compact set [Formula: see text], where [Formula: see text] is a Prufer domain, which has not prime avoidance property. Finally, we show that if [Formula: see text] are valuation domains for a field [Formula: see text] and [Formula: see text] for some [Formula: see text], then there exists [Formula: see text] such that [Formula: see text].

Some topology on zero-dimensional subrings of product of rings

Let R be a ring and {Ri}i?I a family of zero-dimensional rings. We define the Zariski topology on Z(R,?Ri) and study their basic properties. Moreover, we define a topology on Z(R,?Ri) by using ultrafilters; it is called the ultrafilter topology and we demonstrate that this topology is finer than the Zariski topology. We show that the ultrafilter limit point of a collections of subrings of Z(R,?Ri) is a zero-dimensional ring. Its relationship with F-lim and the direct limit of a family of rings are studied.

Study on structure and properties of two metal coordination polymers prepared by 3,5-Bis (4-carboxy-phenoxy)benzoic acid

Two new metal-organic frameworks (MOFs) [Zn(HBCPBA)(tpim)]·H2O (1) and [Co(HBCPBA)(tpim)] (2) were achieved by reactions of corresponding metal salt with mixed organic ligand of 3,5-Bis(4-carboxy-phenoxy)benzoic acid(H3BCPBA) and 2,4,5-tris(4-pyridyl)imidazole (tpim). These crystals were clearly characterized by elemental analysis, IR spectra, X-ray powder diffraction (PXRD) as well as X-ray single-crystal diffraction. The crystal structures showed that the compound 1 crystallizes in the monoclinic system, with P21/c space group, displays a two-dimensional network structure, the compound 2 crystallizes in the orthorhombic system, with Pbca space group, also exhibits a two-dimensional network structure, which has a zero-dimensional ring structure link Co(II) ion by N-containing ligand tpim. Remarkably, 1 can act as fluorescence probe for sensing Fe3+ ion and 2 has electrochemically active substances with good electrochemical reversibility.

Strongly prime ideals and strongly zero-dimensional rings

A prime ideal [Formula: see text] is said to be strongly prime if whenever [Formula: see text] contains an intersection of ideals, [Formula: see text] contains one of the ideals in the intersection. A commutative ring with this property for every prime ideal is called strongly zero-dimensional. Some equivalent conditions are given and it is proved that a zero-dimensional ring is strongly zero-dimensional if and only if the ring is quasi-semi-local. A ring is called strongly [Formula: see text]-regular if in each ideal [Formula: see text], there is an element [Formula: see text] such that [Formula: see text] for all [Formula: see text]. Connections between the concepts strongly zero-dimensional and strongly [Formula: see text]-regular are considered.

Rings all of whose prime serial modules are serial

It is well known that the concept of left serial ring is a Morita invariant property and a theorem due to Nakayama and Skornyakov states that “for a ring [Formula: see text], all left [Formula: see text]-modules are serial if and only if [Formula: see text] is an Artinian serial ring”. Most recently the notions of “prime uniserial modules” and “prime serial modules” have been introduced and studied by Behboodi and Fazelpour in [Prime uniserial modules and rings, submitted; Noetherian rings whose modules are prime serial, Algebras and Represent. Theory 19(4) (2016) 11 pp]. An [Formula: see text]-module [Formula: see text] is called prime uniserial ( [Formula: see text]-uniserial) if its prime submodules are linearly ordered with respect to inclusion, and an [Formula: see text]-module [Formula: see text] is called prime serial ( [Formula: see text]-serial) if [Formula: see text] is a direct sum of [Formula: see text]-uniserial modules. In this paper, it is shown that the [Formula: see text]-serial property is a Morita invariant property. Also, we study what happens if, in the above Nakayama–Skornyakov Theorem, instead of considering rings for which all modules are serial, we consider rings for which every [Formula: see text]-serial module is serial. Let [Formula: see text] be Morita equivalent to a commutative ring [Formula: see text]. It is shown that every [Formula: see text]-uniserial left [Formula: see text]-module is uniserial if and only if [Formula: see text] is a zero-dimensional arithmetic ring with [Formula: see text] T-nilpotent. Moreover, if [Formula: see text] is Noetherian, then every [Formula: see text]-serial left [Formula: see text]-module is serial if and only if [Formula: see text] is serial ring with dim[Formula: see text].

The radical-annihilator monoid of a ring

ABSTRACTKuratowski’s closure-complement problem gives rise to a monoid generated by the closure and complement operations. Consideration of this monoid yielded an interesting classification of topological spaces, and subsequent decades saw further exploration using other set operations. This article is an exploration of a natural analogue in ring theory: a monoid produced by “radical” and “annihilator” maps on the set of ideals of a ring. We succeed in characterizing semiprime rings and commutative dual rings by their radical-annihilator monoids, and we determine the monoids for commutative local zero-dimensional (in the sense of Krull dimension) rings.

When is the annihilating ideal graph of a zero-dimensional semiquasilocal commutative ring planar? Nonquasilocal case

The rings considered in this article are commutative with identity and which are not integral domains. The aim of this article is to classify zero-dimensional semiquasilocal rings with at least two maximal ideals such that their annihilating ideal graphs are planar.

Products of zero-dimensional pairs

Given \(\{R_{i}\}_{i\in I}\) and \(\{S_{i}\}_{i\in I}\) two families of zero-dimensional rings such that \(R_{i}\subseteq S_{i}\) for each \(i\in I\). This paper deals with the transfer of the notion “being a directed union of Artinian subrings” from \(\prod _{i\in I}R_{i}\) to \(\prod _{i\in I}S_{i}\).

Hydrothermal Synthesis and Structure Characterization of Two Hybrid Metal Phosphonates Materials

Under hydrothermal conditions, two copper phosphonates, [Cu2(H2O)2(H2L)(bpy)2(H3L)2]·2H2O (1) and [Cu2(H2O)2(H2L)2(phen)2]·6H2O (2) (H4L = p-xylylenediphosphonic acid, bpy = 2,2'-bipyridine and phen = 1,10-phenanthroline), have been synthesized using diethyl p-xylylenediphosphonate (dixdp), in which p-xylylenediphosphonic acid (H4L) was generated via in situ hydrolysis. Complexe 1 forms a zero-dimensional (0D) bimetallic rings, while complex 2 features a 0D structure containing two kinds of partially deprotonated H3L- and H2L2- ligand.

When is the annihilating ideal graph of a zero-dimensional quasisemilocal commutative ring complemented?

Let R be a commutative ring with identity. Let A(R) denote the collection of all annihilating ideals of R (that is, A(R) is the collection of all ideals I of R which admits a nonzero annihilator in R). Let AG(R) denote the annihilating ideal graph of R. In this article, necessary and sufficient conditions are determined in order that AG(R) is complemented under the assumption that R is a zero-dimensional quasisemilocal ring which admits at least two nonzero annihilating ideals and as a corollary we determine finite rings R such that AG(R) is complemented under the assumption that A(R) contains at least two nonzero ideals.

Commutative rings with infinitely many maximal subrings

It is shown that RgMax (R) is infinite for certain commutative rings, where RgMax (R) denotes the set of all maximal subrings of a ring R. It is observed that whenever R is a ring and D is a UFD subring of R, then | RgMax (R)| ≥ | Irr (D) ∩ U(R)|, where Irr (D) is the set of all non-associate irreducible elements of D and U(R) is the set of all units of R. It is shown that every ring R is either Hilbert or | RgMax (R)| ≥ ℵ0. It is proved that if R is a zero-dimensional (or semilocal) ring with | RgMax (R)| < ℵ0, then R has nonzero characteristic, say n, and R is integral over ℤn. In particular, it is shown that if R is an uncountable artinian ring, then | RgMax (R)| ≥ |R|. It is observed that if R is a noetherian ring with |R| > 2ℵ0, then | RgMax (R)| ≥ 2ℵ0. We determine exactly when a direct product of rings has only finitely many maximal subrings. In particular, it is proved that if a semisimple ring R has only finitely many maximal subrings, then every descending chain ⋯ ⊂ R2 ⊂ R1 ⊂ R0 = R where each Ri is a maximal subring of Ri-1, i ≥ 1, is finite and the last terms of all these chains (possibly with different lengths) are isomorphic to a fixed ring, say S, which is unique (up to isomorphism) with respect to the property that R is finitely generated as an S-module.

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.

Hereditary Properties between a Ring and its Maximal Subrings

We study the existence of maximal subrings and hereditary properties between a ring and its maximal subrings. Some new techniques for establishing the existence of maximal subrings are presented. It is shown that if R is an integral domain and S is a maximal subring of R, then the relation dim(R) = 1 implies that dim(S) = 1 and vice versa if and only if (S : R) = 0. Thus, it is shown that if S is a maximal subring of a Dedekind domain R integrally closed in R; then S is a Dedekind domain if and only if S is Noetherian and (S : R) = 0. We also give some properties of maximal subrings of one-dimensional valuation domains and zero-dimensional rings. Some other hereditary properties, such as semiprimarity, semisimplicity, and regularity are also studied.

On Maximal Subrings of Commutative Rings

A proper subring S of a ring R is said to be maximal if there is no subring of R properly between S and R. If R is a noetherian domain with |R| > 2ℵ0, then | Max (R)| ≤ | RgMax (R)|, where RgMax (R) is the set of maximal subrings of R. A useful criterion for the existence of maximal subrings in any ring R is also given. It is observed that if S is a maximal subring of a ring R, then S is artinian if and only if R is artinian and integral over S. Surprisingly, it is shown that any infinite direct product of rings has always maximal subrings. Finally, maximal subrings of zero-dimensional rings are also investigated.

Of Chains and Rings: Synthetic Strategies and Theoretical Investigations for Tuning the Structure of Silver Coordination Compounds and Their Applications

Varying the polyethyleneglycol spacer between two (iso)-nicotinic groups of the ligand systems, a large structural variety of silver coordination compounds was obtained, starting with zero-dimensional ring systems, via one-dimensional chains, helices and double-helices to two-dimensional polycatenanes. Theoretical calculations help to understand their formation and allow predictions in some cases. These structures can be tuned by careful design of the ligand, the use of solvent and the counter ions, influencing also other important properties such as light stability and solubility. The latter is important in the context of biomedical applications, using silver compounds as antimicrobial agents.

A Nagata-like theorem for certain function spaces

Let X be a space, and let A be a zero-dimensional topological ring. In this paper we will consider a few natural questions that arise when studying the space Cp(X, A), the ring of continuous functions from X to A, endowed with the topology of pointwise convergence. It will be shown that the zero-dimensionality of the codomain plays a vital role in this study. An upper and lower bound will be determined for the density of Cp(X, A) using the density of A and the weight of X. The character of Cp(X, A) will be computed, thus characterizing when Cp(X, A) is metrizable. Lastly, we will consider the topological dual space of Cp(X, A) and use it to prove a Nagata-like theorem.

ARTINIAN SUBRINGS OF PRODUCT OF ZERO-DIMENSIONAL RINGS#

ABSTRACT Given a family of commutative rings {R α}α∈A , we investigate the family 𝒜(∏α∈A R α) of Artinian subrings of ∏α∈A R α. This paper deals also with the question of when a pair (∏α∈A R α, ∏α∈A T α) is zero-dimensional, where {(R α, T α)}α∈A is a family of zero-dimensional pairs.

Clean Semiprime f-Rings with Bounded Inversion

An element in a ring is called clean if it may be written as a sum of a unit and idempotent. The ring itself is called clean if every element is clean. Recently,Anderson and Camillo (Anderson,D. D.,Camillo,V. (2002). Commutative rings whose elements are a sum of a unit and an idempotent. Comm. Algebra 30(7):3327–3336) has shown that for commutative rings every von-Neumann regular ring as well as zero-dimensional rings are clean. Moreover,every clean ring is a pm-ring,that is every prime ideal is contained in a unique maximal ideal. In the same article,the authors give an example of a commutative ring which is a pm-ring yet not clean,e.g.,C(ℝ). It is this example which interests us. Our discussion shall take place in a more general setting. We assume that all rings are commutative with 1.