Space Of Minimal Prime Ideals Research Articles

Zip rings (resp. czip rings) and some applications

A commutative ring with identity is called a zip ring if each of its faithful ideals contains a finitely generated faithful ideal. A natural generalization of this notion is to require every faithful ideal to contain a countably generated faithful ideal. This is what we do in this paper. If a ring has this property, we call it a czip ring. Clearly, every zip ring is a czip ring. We give several characterizations of czip rings, including some in terms of subspaces of the space of minimal prime ideals with the Zariski topology. In rings of continuous functions we characterize the Tychonoff spaces X for which C(X) is a czip ring. This enables us to show that the class of czip rings properly contains that of zip rings.

On some properties of the space of minimal prime ideals of 𝐶𝑐 (𝑋)

In this article we consider some relations between the topological properties of the spaces X and Min(Cc (X)) with algebraic properties of Cc (X). We observe that the compactness of Min(Cc (X)) is equivalent to the von-Neumann regularity of qc (X), the classical ring of quotients of Cc (X). Furthermore, we show that if 𝑋 is a strongly zero-dimensional space, then each contraction of a minimal prime ideal of 𝐶(𝑋) is a minimal prime ideal of Cc(X) and in this case 𝑀𝑖𝑛(𝐶(𝑋)) and Min(Cc (X)) are homeomorphic spaces. We also observe that if 𝑋 is an Fc-space, then Min(Cc (X)) is compact if and only if 𝑋 is countably basically disconnected if and only if Min(Cc(X)) is homeomorphic with β0X. Finally, by introducing zoc-ideals, countably cozero complemented spaces, we obtain some conditions on X for which Min(Cc (X)) becomes compact, basically disconnected and extremally disconnected.

Zero-divisor graph of the rings Cp(x) and Cp∞ (x)

In this article we introduce the zero-divisor graphs ?P(X) and ?P? (X) of the two rings CP(X) and CP? (X); here P is an ideal of closed sets in X and CP(X) is the aggregate of those functions in C(X), whose support lie on P. CP? (X) is the P analogue of the ring C?(X). We determine when the weakly zero-divisor graph W?P(X) of CP(X) coincides with ?P(X). We find out conditions on the topology on X, under-which ?P(X) (respectively, ?P? (X)) becomes triangulated/ hypertriangulated. We realize that ?P(X) (respectively, ?P? (X)) is a complemented graph if and only if the space of minimal prime ideals in CP(X) (respectively ?P? (X)) is compact. This places a special case of this result with the choice P ? the ideals of closed sets in X, obtained by Azarpanah and Motamedi in [8] on a wider setting. We also give an example of a non-locally finite graph having finite chromatic number. Finally it is established with some special choices of the ideals P and Q on X and Y respectively that the rings CP(X) and CQ(Y) are isomorphic if and only if ?P(X) and ?Q(Y) are isomorphic.

Zero-divisor graph of real-valued continuous functions on a frame

The main object of this paper is to study the zero-divisor graph Г(RL) of the ring RL. Using the properties of the lattice Coz L, we associate the ring properties of RL, the graph properties of Г(RL), and the properties of a competely regular frame L. Paths in Г(RL) are investigated, and it is shown that the diameter of Г(RL) and the girth of Г(RL) coincide whenever L has at least 5 elements. Cycles in Г(RL) are surveyed, a ring-theoretic and a frame-theoretic characterizations are provided for the graph Г(RL) to be triangulated or be hypertriangulated. We show that Г(RL) is complemented if and only if the space of minimal prime ideals of RL is compact. The relation between the clique number of Г(RL), the cellularity of L and the dominating number of Г(RL) is given. Finally, we prove that if Г(RL) is not triangulated, then the set of centers of Г(RL) is a dominating set if and only if the socle of RL is an essential ideal.

The hull-kernel topology on prime ideals in posets

In this paper, we continue our study of prime ideals in posets that was started in Joshi and Mundlik (Cent Eur J Math 11(5):940---955, 2013) and, Erne and Joshi (Discrete Math 338:954---971, 2015). We study the hull-kernel topology on the set of all prime ideals $$\mathcal {P}(Q)$$P(Q), minimal prime ideals $$\mathrm{Min}(Q)$$Min(Q) and maximal ideals $$\mathrm{Max}(Q)$$Max(Q) of a poset Q. Then topological properties like compactness, connectedness and separation axioms of $$\mathcal {P}(Q)$$P(Q) are studied. Further, we focus on the space of minimal prime ideals $$\mathrm{Min}(Q)$$Min(Q) of a poset Q. Under the additional assumption that every maximal ideal is prime, the collection of all maximal ideals $$\mathrm{Max}(Q)$$Max(Q) of a poset Q forms a subspace of $$\mathcal {P}(Q)$$P(Q). Finally, we prove a characterization of a space of maximal ideals of a poset to be a normal space.

The minimal prime spectrum of rings with annihilator conditions

In this paper, we study rings with the annihilator condition (a.c.) and rings whose space of minimal prime ideals, Min ( R ) , is compact. We begin by extending the definition of (a.c.) to noncommutative rings. We then show that several extensions over semiprime rings have (a.c.). Moreover, we investigate the annihilator condition under the formation of matrix rings and classical quotient rings. Finally, we prove that if R is a reduced ring then: the classical right quotient ring Q ( R ) is strongly regular if and only if R has a Property (A) and Min ( R ) is compact, if and only if R has (a.c.) and Min ( R ) is compact. This extends several results about commutative rings with (a.c.) to the noncommutative setting.

Rings with Property (A) and their extensions

A commutative ring R has Property (A) if every finitely generated ideal of R consisting entirely of zero-divisors has a nonzero annihilator. We continue in this paper the study of rings with Property (A). We extend Property (A) to noncommutative rings, and study such rings. Moreover, we study several extensions of rings with Property (A) including matrix rings, polynomial rings, power series rings and classical quotient rings. Finally, we characterize when the space of minimal prime ideals of rings with Property (A) is compact.

Cozero complemented spaces; when the space of minimal prime ideals of a C( X) is compact

If X is a Tychonoff space, C( X) its ring of real-valued continuous functions, and f∈ C( X), then the cozeroset of f is coz( f)= {x∈X: f(x)≠0} . If, for every cozeroset V of X, there is a disjoint cozeroset V′ such that V∪ V′ is dense in X, then X is said to be cozero complemented. It has long been known that X is cozero complemented iff the space Min C( X) of minimal prime ideals of C( X) (in the hull-kernel or Zariski topology) is compact iff the classical ring of fractions of C( X) is von Neumann regular. While many characterizations of cozero complemented spaces are known, they seem not to be adequate to answer some natural questions about them raised by R. Levy and J. Shapiro in an unpublished preprint. These questions concern the relationship between a space being cozero complemented and certain kinds of subspaces having this property, and between a product of two spaces being cozero complemented and the factor spaces being cozero complemented. Also, some conditions are given that guarantee that a space that is locally cozero complemented has this property globally. In this paper partial answers are given to these questions. Sample results: If X is weakly Lindelöf and dense in T, then X is cozero complemented iff T is cozero complemented; if X× Y is weakly Lindelöf and cozero complemented, then X and Y are cozero complemented, but if D is an uncountable discrete space, then βD× βD is not cozero complemented even though βD is cozero complemented. If X is locally cozero complemented and either weakly Lindelöf or paracompact, then X is cozero complemented.

A Pointfree Study of Bases For Spaces Of Minimal Prime Ideals

Let C (X) denote the ring of all real-valued continuous functions on a topological space X, and mX its space of minimal prime ideals with the hull-kernel topology. It has a base consisting of clopen sets and which contains the closure of the union of any sequence of its members. In the paper Spaces with a pretty base ( J. Pure and Appl. Algebra 70 (1991), 81-87), spaces with a base with these properties are studied and are shown to have almost all of the known properties of the space mX including the fact that if it is a weakly Lindel¨ of space, then it is basically disconnected. In the present paper, analogous results are derived in a pointfree context in which topological spaces are replaced by frames. In some cases, we are able to obtain more general results with simpler proofs.

The global dimension of an f-ring via its space of minimal prime ideals

This article examines hereditary f-rings, after first characterizingthe hereditary von Neumann regular commutative rings as those for which the space of maximal ideals is hereditarily paracompact. It is shown that if C ( X , Z ) , the ring of integer-valued continuous functions on a zero-dimensional space X, is hereditary, then X is finite. This is shown two ways; once as a consequence of the following: if A is any singular archimedean f-ring, then A/2A is a boolean ring, and gd(A) >= gd(A/2A)+ 1 , where gd(A) stands for the global dimension of A. As a consequence of this it is also shown that if A is a singular archimedean f-ring and gd(A) <= 2, then Min(A), the space of minimal prime ideals is hereditarily paracompact. The paper concludes with a calculation of the global dimension of a mihereditary singular archimedean f-ring A, in which the cellularityof Min(A) is “much less” than |A| if finite, it is k+2, where Nk = |A|.

Minimal primal ideals in Banach algebras

AbstractAn ideal I in a ring R is primal if whenever J1, …, Jn is a finite set of ideals of R with J1 … Jn = {0} then Ji ⊆ I for at least one i ∈ {1, …, n}. If the ring is commutative then is is easily shown that each primal ideal contains a prime ideal. In this paper it is shown that in a separable semi-simple Banach algebra each primal ideal contains a prime ideal, and that the space of minimal prime ideals is compact and extremally disconnected in the hull-kernel topology. An example is given of an inseparable C-algebra with a primal ideal which does not contain a prime ideal.

The Prime Spectrum of an MV‐Algebra

AbstractIn this paper we show that the prime ideal space of an MV‐algebra is the disjoint union of prime ideal spaces of suitable local MV‐algebras. Some special classes of algebras are defined and their spaces are investigated. The space of minimal prime ideals is studied as well.Mathematics Subject Classification: 03B50, 06D99.

Fraction-Dense Algebras and Spaces

AbstractA fraction-dense (semi-prime) commutative ring A with 1 is one for which the classical quotient ring is rigid in its maximal quotient ring. The fractiondense ƒ- rings are characterized as those for which the space of minimal prime ideals is compact and extremally disconnected. For archimedean lattice-ordered groups with this property it is shown that the Dedekind and order completion coincide. Fractiondense spaces are defined as those for which C(X) is fraction-dense. If X is compact, then this notion is equivalent to the coincidence of the absolute of X and its quasi-F cover. R-embeddings of Tychonoff spaces are re-introduced and examined in the context of fraction-density.

The Space of Minimal Prime Ideals of C(X) Need Not Be Basically Disconnected

The Space of Minimal Prime Ideals of C(X) Need Not Be Basically Disconnected

The space of minimal prime ideals of 𝐶(𝑋) need not be basically disconnected

Problems posed twenty and twenty-five years ago by M. Henriksen and M. Jerison are solved by showing that the space of minimal prime ideals of the ring C ( X ) C\left ( X \right ) of continuous real-valued functions on a compact (Hausdorff) space need not be basically disconnected—or even an F F -space.

Maximal d-Ideals in a Riesz Space

We recall that the ideal I in an Archimedean Riesz space L is called a d-ideal whenever it follows from ƒ ∊ I that {ƒ}dd ⊂ I. Several authors (see [4], [5], [6], [12], [13], [15] and [18]) have considered the class of all d-ideals in L, but the set ℐd of all maximal d-ideals in L has not been studied in detail in the literature. In [12] and [13] the present authors paid some attention to certain aspects of the theory of maximal d-ideals, however neglecting the fact thatℐd, equipped with its hull-kernel topology, is a structure space of the underlying Riesz space L.The main purpose of the present paper is to investigate the topological properties of ℐd and to compare ℐd to other structure spaces of L, such as the space of minimal prime ideals and the space of all e-maximal ideals in L (where e &gt; 0 is a weak order unit).

Space of minimal prime ideals of a ring without nilpotent elements

Hull-kernel topology on the set ∑(R) of prime ideals of a ring R with unity and without nilpotent elements is discussed. The restriction of this topology to the set π( R) of minimal prime ideals of R has been investigated in detail. The compactness of π( R) has been characterized in several ways. An interesting characterization of Baer rings is given. A functorial correspondence between the category of rings having the property that every prime ideal contains a unique minimal prime ideal and their minimal spectra is established.

The space of minimal prime ideals in a 0-distributive semilattice

LetS be a 0-distributive semilattice and\(\mathfrak{M}\) be its minimal spectrum. It is shown that\(\mathfrak{M}\) is Hausdorff. The compactness of\(\mathfrak{M}\) has been characterized in several ways. A representation theorem (like Stone's theorem for Boolean algebras) for disjunctive, 0-distributive semilattices is obtained.

Stability in Witt rings

An abstract Witt ring R is defined to be a certain quotient of an integral group ring for a group of exponent 2. The ring R has a unique maximal ideal M containing 2. A variety of results are obtained concerning n-stability, the condition that M n + 1 = 2 M n {M^{n + 1}} = 2{M^n} , especially its relationship to the ring of continuous functions from the space of minimal prime ideals of R to the integers. For finite groups, a characterization of integral group rings is obtained in terms of n-stability. For Witt rings of formally real fields, conditions equivalent to n-stability are given in terms of the real places defined on the field.

The Minimal Prime Spectrum of a Commutative Ring

We call a topological space X minspectral if it is homeomorphic to the space of minimal prime ideals of a commutative ring A in the usual (hull-kernel or Zariski) topology (see [2, p. 111]). Note that if A has an identity, is a subspace of Spec A (as defined in [1, p. 124]). It is well known that a minspectral space is Hausdorff and has a clopen basis (and hence is completely regular). We give here a topological characterization of the minspectral spaces, and we show that all minspectral spaces can actually be obtained from rings with identity and that open (but not closed) subspaces of minspectral spaces are minspectral (Theorem 1, Proposition 5).