Maximal Ideals Research Articles

Nagata-like theorems for integral domains of finite character and finite t-character

An integral domain D is said to be of finite character (resp., finite t-character) if every nonzero nonunit of D belongs to at most a finite number of maximal ideals (resp., maximal t-ideals) of D. Let S be a multiplicative set of D. In this paper we study when DSbeing of finite character (resp., finite t-character) implies that D is of finite character (resp., finite t-character).

Finiteness and Skolem Closure of Ideals for Non-Unibranched Domains

A one-dimensional, Noetherian, local domain D with maximal ideal 𝔪 and finite residue field was known to be an almost strong Skolem ring if analytically irreducible. It was unknown whether this condition is necessary. We show that it is at least necessary for D to be unibranched. After introducing a general notion of equalizing ideal, we show that, for k large enough, the ideals of the form 𝔐 k, a = {f ∈ Int(D) | f(a) ∈ 𝔪 k }, for a ∈ D, are distinct. This allows to show that the maximal ideals 𝔐 a = {f ∈ Int(D) | f(a) ∈ 𝔪}, although not necessarily distinct, are never finitely generated.

Direct products of modules whose endomorphism rings have at most two maximal ideals

Monogeny classes and epigeny classes have proved to be useful in the study of direct sums of uniserial modules and other classes of modules. In this paper, we show that they also turn out to be useful in the study of direct products.

A note on ideals of [formula omitted

For a topological space X, let Σ(X) be the family of all subrings of C(X) containing C⁎(X). In 1997, S.K. Acharyya, K.C. Chattopadhyay and D.P. Ghosh showed that for an A(X)∈Σ(X), the intersection of all free maximal ideals of A(X) is an ideal of C∞(X) containing CK(X). In this paper we define z∞-ideals and nicely balanced ideals of C∞(X) and show that if I is a z∞-ideal of C∞(X) then I is the intersection of all free maximal ideals of A(X) for some A(X)∈Σ(X) if and only if I is nicely balanced. Examples related to z∞-ideals and nicely balanced ideals are given.

Maximal Left Ideals In Local Goldie (-1, 1) Rings

In this paper, we show how to reduce the study of nondegenerate local Goldie (-1, 1) rings to the strongly prime case, via the notions of uniform ideals and essential subdirect product. Also, we construct the maximal left quotient ring of (-1, 1) ring that is a left quotient ring of itself. We follow Utumi where a maximal left quotient ring is constructed as a direct limit of a partially defined homomorphism from the left ideal of R to R.

One-sided ideals of the non-commutative Schwartz space

We show that maximal one-sided ideals of the non-commutative Schwartz space are closed. We also characterize all closed one-sided ideals of this algebra. As a result, all maximal left ideals are fixed.

Maximal, Irreducible and Prime Soft Ideals of BCK/BCI-Algebra

In this paper, the notions of soft irreducible, prime and maximal soft ideals,irreducible, prime and maximal soft idealistic over on BCK/BCIalgebras are introduced, and several examples are given to illustrate. Relations between irreducible, prime and maximal idealistic soft BCK/BCI-algebras are investigated. 2000 AMS Classification: 06D72; 06F35; 46J20.

On maximal ideals in certain reduced twisted C*-crossed products

AbstractWe consider a twisted action of a discrete groupGon a unital C*-algebraAand give conditions ensuring that there is a bijective correspondence between the maximal invariant ideals ofAand the maximal ideals in the associated reduced C*-crossed product.

A characterization of modules embedding in products of primes and enveloping condition for their class

Modules (cogenerated by nonzero their submodules) having nonzero square homomorphism in nonzero submodules are said (prime) weakly compressible (wc). Such modules are semiprime (i.e. they are cogenerated by their essential submodules). For many rings R, including commutative rings, it is proved that wc modules are isomorphic submodules of products of prime modules, and the converse holds precisely when the class of wc modules is enveloping for mod-R or equivalently, every semiprime module is wc. Semi-Artinian rings R have the latter property and the converse is true when R is strongly regular. Duo Noetherian rings over which wc modules form an enveloping class are shown to have a finite number maximal ideals. If R is Morita equivalent to a Dedekind domain, then the class of wc R-modules is enveloping if and only if R is simple Artinian or J (R) ≠ 0.

Prime and Maximal Ideals in Ternary Semigroups

This paper deals with prime and maximal ideals in ternary semigroup. In this paper we shall study the intersection of all prime ideals and intersection of all maximal ideals in a (non-commutative) ternary semigroup T. In particular we give a rather general necessary and sufficient condition in order that the set of all maximal ideals coincides with the set of all prime ideals.

$r$-ideals in commutative rings

In this article we introduce the concept of $r$-ideals in commutative rings (note: an ideal $I$ of a ring $R$ is called $r$-ideal, if $ab\in I$ and ${\rm Ann}(a)=(0)$ imply that $b\in I$ for each $a,b\in R$). We study and investigate the behavior of $r$-ideals and compare them with other classical ideals, such as prime and maximal ideals. We also show that some known ideals such as $z^\circ$-ideals are $r$-ideals. It is observed that if $I$ is an $r$-ideal, then so too is a minimal prime ideal of $I$. We naturally extend the celebrated results such as Cohen's theorem for prime ideals and the Prime Avoidance Lemma to $r$-ideals. Consequently, we obtain interesting new facts related to the Prime Avoidance Lemma. It is also shown that $R$ satisfies property $A$ (note: a ring $R$ satisfies property $A$ if each finitely generated ideal consisting entirely of zerodivisors has a nonzero annihilator) if and only if for every $r$-ideal $I$ of $R$, $I[x]$ is an $r$-ideal in $R[x]$. Using this concept in the context of $C(X)$, we show that every $r$-ideal is a $z^\circ$-ideal if and only if $X$ is a $\partial$-space (a space in which the boundary of any zeroset is contained in a zeroset with empty interior). Finally, we observe that, although the socle of $C(X)$ is never a prime ideal in $C(X)$, the socle of any reduced ring is always an $r$-ideal.

Maximal ideals in some F-algebras of holomorphic functions

For 1 < p < ?, the Privalov class Np consists of all holomorphic functions f on the open unit disk D of the complex plane C such that sup 0?r<1?2?0 (log+ |f(rei?)j|p d?/2? < + ? M. Stoll [16] showed that the space Np with the topology given by the metric dp defined as dp(f,g) = (?2?0 (log(1 + |f*(ei?) - g*(ei?)|))p d?/2?)1/p, f,g ? Np; becomes an F-algebra. Since the map f ? dp(f,0) (f ? Np) is not a norm, Np is not a Banach algebra. Here we investigate the structure of maximal ideals of the algebras Np (1 < p < ?). We also give a complete characterization of multiplicative linear functionals on the spaces Np. As an application, we show that there exists a maximal ideal of Np which is not the kernel of a multiplicative continuous linear functional on Np.

Essential ideals in subrings of C(X) that contain C*(X)

Let A(X) be a subring of C(X) that contains C*(X). In Redlin and Watson (1987) and in Panman et al. (2012), correspondences ZA and ?A are defined between ideals in A(X) and z-filters on X, and it is shown that these extend the well-known correspondences studied separately for C*(X) and C(X), respectively, to any intermediate ring A(X). Moreover, the inverse map Z-1A sets up a one-one correspondence between the maximal ideals of A(X) and the z-ultrafilters on X. In this paper, first, we characterize essential ideals in A(X). Afterwards, we show that Z-1A maps essential (resp., free) z-filters on X to essential (resp., free) ideals in A(X) and Z-1A maps essential ?A-filters to essential ideals. Similar to C(X) we observe that the intersection of all essential minimal prime ideals in A(X) is equal to the socle of A(X). Finally, we give a new characterization for the intersection of all essential maximal ideals of A(X).

Strongly fixed ideals in $ C (L)$ and compact frames

Let $C(L)$ be the ring of real-valued continuous functions on a frame $L$. In this paper, strongly fixed ideals and characterization of maximal ideals of $C(L)$ which is used with strongly fixed are introduced. In the case of weakly spatial frames this characterization is equivalent to the compactness of frames. Besides, the relation of the two concepts, fixed and strongly fixed ideals of $C(L)$, is studied particularly in the case of weakly spatial frames. The concept of weakly spatiality is actually weaker than spatiality and they are equivalent in the case of conjunctive frames. Assuming Axiom of Choice, compact frames are weakly spatial.

Types of fuzzy ideals in fuzzy lattices

In this paper we consider the notion of Fuzzy Lattices, which was introduced by Chon (Korean J. Math 17 (2009), No. 4, 361-374). We propose some new notions for Fuzzy Ideals and Filters and provide a characterization of Fuzzy Ideals via α-level Sets and Support. Some types of ideals and filters, such as: Fuzzy Principal Ideals (Filters), Proper Fuzzy Ideals (Filters), Prime Fuzzy Ideals (Filters) and Fuzzy Maximal Ideals (Filters) are also provided. Some properties (analogous to the classical theory) are also proved and the notion of Homomorphism from fuzzy lattices as well as the demonstration of some important propositions about it are also provided.

The Set of Intermediate Rings as a Boolean Algebra

Let R ⊂ S be an extension of integral domains, Supp(S/R) = {P ∈ Spec(R): (S/R) P ≠ 0} the support of the R-module S/R, and [R, S] the set of intermediate rings between R and S. We prove the following theorem: If there is a maximal chain of rings from R to S of length n and Supp(S/R) consists of n maximal ideals, then ([R, S], ·, ∩) is a boolean algebra with cardinality 2 n . An interesting example is exhibited to illustrate the scope of this theorem.

Simple weight modules over weak generalized Weyl algebras

In this paper we address the problem of classification of simple weight modules over weak generalized Weyl algebras of rank one. The principal difference between weak generalized Weyl algebras and generalized Weyl algebras is that weak generalized Weyl algebras are defined using an endomorphism rather than an automorphism of a commutative ring R. We reduce classification of simple weight modules over weak generalized Weyl algebras to description of the dynamics of the action of the above mentioned endomorphism on the set of maximal ideals. We also describe applications of our results to the study of generalized Heisenberg algebras.

On implicative and maximal ideals of BL-algebras

In the theory of MV-algebras, implicative ideals are studied by Hoo and Sessa. In this paper we define and characterize implicative ideals of BL-algebras. We also investigate maximal ideals of BL-algebras and prove that if an ideal is prime and implicative, then it is maximal. Moreover, we show that an ideal is maximal if and only if the quotient BL-algebra is a simple MV-algebra. Finally, we give the homomorphic properties of implicative and maximal ideals.

Concerning maximal $${\ell}$$ ℓ -ideals of rings of continuous integer-valued functions

Maximal \({\ell}\)-ideals of the ring \({C(X, \mathbb{Z})}\) of continuous integer-valued functions on a topological space X were characterised by Subramanian to be exactly the minimal prime ideals of this ring. This note supplements this result by showing that, in fact, these ideals are also exactly the maximal z-ideals, exactly the maximal d-ideals, and exactly the maximal pure ideals of this ring. Casting everything in the category ODFrm of zero-dimensional frames, we show that what has just been said holds in any ring \({\mathfrak{Z}L}\) of continuous integer-valued functions on a zero-dimensional frame L. The ideals in question are describable in terms of the cozero map. They are precisely the inverse images (under the cozero map) of the points of the universal zero-dimensional compactification of L.

Graph properties of co-maximal ideal graphs of commutative rings

In this paper, we determine the diameters of graphs [Formula: see text] and [Formula: see text] for a ring R with infinitely many maximal ideals. We also use graph blow-up to give a complete classification of rings R whose graphs [Formula: see text] are non-empty planar graphs.