Maximal Ideals Research Articles

The cone and cylinder algebras

In this exposition-type note we present detailed proofs of certain assertions concerning several algebraic properties of the cone and cylinder algebras. These include a determination of the maximal ideals, the solution of the B\'ezout equation and a computation of the stable ranks by elementary methods.

On Some Properties of *-annihilators and *-maximal Ideals in Rings with Involution

We describe the ∗-right annihilator (∗-left anihilator) of a subset of a ring and we investigate the relationships between the right annihilator and ∗-right annihilator. These connections permit the transfer of various properties from annihilators to ∗-annihilators . It is known that the quotient ring constructed from a ring and a maximal ideal is a field, whereas we prove that the quotient ring constructed from a ring and a *-maximal ideal is not a *-field. Equivalent definitions to ∗-regular ring are given.

When is $C(X)$ polynomially ideal?

Let $\mathbb{A}$ be a commutative $f$-algebra with unit. The sets of all ideals in $\mathbb{A}$ and all intersections of maximal ideals in $\mathbb{A}$ are denoted by $\mathfrak{I}(\mathbb{A})$ and $\mathfrak{IM}% (\mathbb{A})$, respectively. Whenever $\mathfrak{a}% \in\mathfrak{I}(\mathbb{A})$, we say that $\mathbb{A}$ is polynomially $\mathfrak{a}$-ideal if, for every $f\in\mathbb{A}$ with $p(f)\in\mathfrak{a}$ for some non-zero polynomial $p(x)$, there is an $f_{0}\in\mathfrak{a}$ such that $p(f+f_{0})=0$. We prove that if $\mathbb{A}$ is bounded inversion closed and $\mathfrak{a}\in\mathfrak{IM}(\mathbb{A})$, then $\mathbb{A}$ is polynomially $\mathfrak{a}$-ideal if and only if idempotents lift modulo $\mathfrak{a}$. This fact is based upon a systematic study of idempotent elements of an $f$-algebra. As a consequence, we show that, if $X$ is a Tychonoff space, then $C(X)$ is polynomially $\mathfrak{a}$-ideal for all $\mathfrak{a}\in\mathfrak{I}(C(X))$ if and only if $X$ is a $P$-space. Moreover, we prove that $C(X)$ is polynomially $\mathfrak{a}$-ideal for all $\mathfrak{a}\in\mathfrak{IM}(C(X))$ if and only if $X$ is strongly zero-dimensional. It turns out that this extends a theorem by Miers, namely, if $X$ is a compact Hausdorff space, then $C(X)$ is polynomially $\mathfrak{a}$-ideal for every uniformly closed ideal $\mathfrak{a}$ in $C(X)$ if and only if $X$ is totally disconnected.

Almost uniserial rings and modules

We study the class of almost uniserial rings as a straightforward common generalization of left uniserial rings and left principal ideal domains. A ring R is called almost left uniserial if any two non-isomorphic left ideals of R are linearly ordered by inclusion, i.e., for every pair I, J of left ideals of R either I⊆J, or J⊆I, or I≅J. Also, an R-module M is called almost uniserial if any two non-isomorphic submodules are linearly ordered by inclusion. We give some interesting and useful properties of almost uniserial rings and modules. It is shown that a left almost uniserial ring is either a local ring or its maximal left ideals are cyclic. A Noetherian left almost uniserial ring is a local ring or a principal left ideal ring. Also, a left Artinian principal left ideal ring R is almost left uniserial if and only if R is left uniserial or R=M2(D), where D is a division ring. Finally we consider Artinian commutative rings which are almost uniserial and we obtain a structure theorem for these rings.

On the Topological Stable Rank of a Noncommutative Version of the Disc Algebra

Based on a bounded approximate version of LU decomposition, we calculate the topological stable rank of \({{\mathcal {A}}}_{{\mathcal {N}}}\), which is a noncommutative version of the disc algebra. In addition, a general characterization of the maximal two-sided ideals of \({{\mathcal {A}}}_{{\mathcal {N}}}\) will be obtained.

To the theory of partial semirings of continuous [0,∞]-valued functions

The aim of the paper is to investigate the partial semiring C∞+(X) of all continuous functions on a topological space X that take values in the semiring [0, ∞] with the usual topology. Maximal ideals of the partial semiring C∞+(X) are studied, and fundamental properties of its prime ideals are described.

Minimal inversion complete sets and maximal abelian ideals

Let g be a simple Lie algebra, b a fixed Borel subalgebra, and W the Weyl group of g. In this article, we study a relationship between the maximal abelian ideals of b and the minimal inversion complete sets of W. The latter have been recently defined by Malvenuto et al. (2015) [10].

Supersolvability and the Koszul property of root ideal arrangements

A root ideal arrangement A_I is the set of reflecting hyperplanes corresponding to the roots in an order ideal I of the root poset on the positive roots of a finite crystallographic root system. A characterisation of supersolvable root ideal arrangements is obtained. Namely, A_I is supersolvable if and only if I is chain peelable, meaning that it is possible to reach the empty poset from I by in each step removing a maximal chain which is also an order filter. In particular, supersolvability is preserved under taking subideals. We identify the maximal ideals that correspond to non-supersolvable arrangements. There are essentially two such ideals, one in type D_4 and one in type F_4. By showing that A_I is not line-closed if I contains one of these, we deduce that the Orlik-Solomon algebra OS(A_I) has the Koszul property if and only if A_I is supersolvable.

On Nichols (braided) Lie algebras

We prove (i) Nichols algebra 𝔅(V) of vector space V is finite dimensional if and only if Nichols braided Lie algebra 𝔏(V) is finite dimensional; (ii) if the rank of connected V is 2 and 𝔅(V) is an arithmetic root system, then 𝔅(V) = F ⊕ 𝔏(V); and (iii) if Δ(𝔅(V)) is an arithmetic root system and there does not exist any m-infinity element with puu≠ 1 for any u ∈ D(V), then dim (𝔅(V)) = ∞ if and only if there exists V′, which is twisting equivalent to V, such that dim (𝔏-(V′)) = ∞. Furthermore, we give an estimation of dimensions of Nichols Lie algebras and two examples of Lie algebras which do not have maximal solvable ideals.

Two ideals connected with strong right upper porosity at a point

Let $SP$ be the set of upper strongly porous at $0$ subsets of $\mathbb R^{+}$ and let $\hat I(SP)$ be the intersection of maximal ideals $I \subseteq SP$. Some characteristic properties of sets $E\in\hat I(SP)$ are obtained. It is shown that the ideal generated by the so-called completely strongly porous at $0$ subsets of $\mathbb R^{+}$ is a proper subideal of $\hat I(SP).$

Covering maximal ideals with minimal primes

A ring is a UMP-ring if every maximal ideal in the ring is the union of the minimal prime ideals it contains. Banerjee, Ghosh and Henriksen have characterized Tychonoff spaces X for which C(X) is a UMP-ring. One of the characterizations is that every singleton of β X is what is called a nearly round subset. In this article, we define nearly round quotient maps, and use them to characterize completely regular frames L for which $${\mathcal{R} L}$$ is a UMP-ring. All such frames are almost P-frames, and an Oz-frame is of this kind precisely when it is an almost P-frame. If L is perfectly normal (and hence if L is metrizable), then $${\mathcal{R} L}$$ is a UMP-ring if and only if L is Boolean. If A is a UMP-ring which is a $${\mathbb{Q}}$$ -algebra, then every ideal of A, when viewed as a ring in its own right, is a UMP-ring.

When is [formula omitted

This short article is in fact an erratum to Theorem 2.5 in Ghirati and Taherifar (2014) [6]. In this paper, we prove that J(C(X)/CF(X))=0 or equivalently, the socle CF(X) of C(X) coincides with the intersection of all essential maximal ideals of C(X) if and only if every infinite subset of I(X) contains a closed infinite subset, if and only if every pseudocompact subset of X has at most a finite number of isolated points. This fact shows that part (2) of aforementioned Theorem 2.5 is not correct. Our results in this paper also amend Theorem 5.6 in Azarpanah et al. (2008) [2]. Examples are provided to illustrate and delimit our results.

A short proof of Cartan’s Nullstellensatz for entire functions in $${\mathbb{C}^n}$$ C n

Using the fact that the maximal ideals in the polydisk algebra are given by the kernels of point evaluations, we derive a simple formula that gives a solution to the Bézout equation in the space of all entire functions of several complex variables. Thus a short and easy analytic proof of Cartan’s Nullstellensatz is obtained.

Prime and Maximal Ideals in Ternary Semigroups

Prime and Maximal Ideals in Ternary Semigroups

Maximal non-treed subring of its quotient field

An integral domain R is called maximal non-treed subring of its quotient field, if R is not treed, and every proper overring of R is treed. We do prove that R is a maximal non-treed subring of its quotient field if and only if R is non-treed, local with maximal ideal m and its integral closure \(\overline{R}\) is a semi-local Prufer domain with two maximal ideals M, N such that \(M\cap N=m\), and there is no field lying properly between R / m and \(\overline{R}/M\times \overline{R}/N\). Additional characterizations are settled in terms of pullback rings or minimal overrings.

Boundary and shape of Cohen–Macaulay cone

Let $R$ be a Cohen-Macaulay local domain. In this paper we study the cone of Cohen-Macaulay modules inside the Grothendieck group of finitely generated $R$-modules modulo numerical equivalences, introduced in \cite{CK}. We prove a result about the boundary of this cone for Cohen-Macaulay domain admitting de Jong's alterations, and use it to derive some corollaries on finiteness of isomorphism classes of maximal Cohen-Macaulay ideals. Finally, we explicitly compute the Cohen-Macaulay cone for certain isolated hypersurface singularities defined by $\xi\eta - f(x_1, \ldots, x_n)$.

The Krull dimension of power series rings over almost Dedekind domains

Let D be an almost Dedekind domain that is not Dedekind, M be a non-invertible maximal ideal of D, X be an indeterminate over D, and D〚X〛 be the power series ring over D. We first construct an example of η1-sets and we then use this η1-set and M to give a simple proof of dim⁡(D〚X〛)≥2ℵ1. We show that ht(M〚X〛/MD〚X〛)≥2ℵ1 when D has only countably many non-invertible maximal ideals or when M is countably generated. We finally construct a simple example of almost Dedekind domains that are not Dedekind with given cardinal number of non-invertible maximal ideals.

Some results on quasi-unmixed local domains

This paper shows that if (R,m) is a Nœtherian unibranch local domain with field of fractions K, then the integral closure S of R in K is analytically irreducible and finite over R if and only if R is analytically irreducible. We also prove the equality between the number of minimal prime ideals in Rˆ and the number of maximal ideals in S in the case when R is a Nœtherian quasi-unmixed local domain such that S is finite over R and Snˆ has only one minimal prime for all the maximal ideals n in S, where Snˆ is the n-adic completion of Sn.

The maximal two-sided ideals of nest algebras

We give a necessary and sufficient criterion for an operator in a nest algebra to belong to a proper two-sided ideal of that algebra. Using this result, we describe the strong radical of a nest algebra, and give a general description of the maximal two-sided ideals.

INSERTION-OF-FACTORS-PROPERTY WITH FACTORS MAXIMAL IDEALS

Insertion-of-factors-property, which was introduced by Bell, has a role in the study of various sorts of zero-divisors in noncommutative rings. We in this note consider this property in the case that factors are restricted to maximal ideals. A ring is called IMIP when it satisfies such property. It is shown that the Dorroh extension of A by K is an IMIP ring if and only if A is an IFP ring without identity, where A is a nil algebra over a field K. The structure of an IMIP ring is studied in relation to various kinds of rings which have roles in noncommutative ring theory.