Related papers: Simple Cuntz-Pimsner rings
We show that if $A$ is a unital $C^*$-algebra and $B$ is a Cuntz-Krieger algebra for which $A\otimes\mathbb{K} \cong B\otimes\mathbb{K}$, then $A$ is a Cuntz-Krieger algebra. Consequently, corners of Cuntz-Krieger algebras are Cuntz-Krieger…
Pimsner introduced the C*-algebra O_X generated by a Hilbert bimodule X over a C*-algebra A. We look for additional conditions that X should satisfy in order to study simplicity and, more generally, the ideal structure of O_X when X is…
We study the KMS states on local quantum Cuntz-Krieger algebras associated to quantum graphs. Using their isomorphism to the Cuntz-Pimsner algebra of the quantum edge correspondence, we show that the general criteria for KMS states can be…
It is proved that the ring $R$ with center $Z(R)$, such that the module $R_{Z(R)}$ is an essential extension of the module $Z(R)_{Z(R)}$, is not necessarily right quasi-invariant, i.e., maximal right ideals of the ring $R$ are not…
Let $R$ be a commutative ring. When is a subgroup of $(R, +)$ an ideal of $R$? We investigate this problem for the rings $\mathbb{Z}^{d}$ and $\prod_{i=1}^{d} \mathbb{Z}_{n_{i}}$. For various subgroups of these rings we obtain necessary and…
The theory of Gromov-Hausdorff convergence is applied to sequences of quotient rings of integers. It is shown the existence of limit rings (fields) as the Gromov-Hausdorff limits of sequences of metric quotient rings. The relation of these…
In this paper we introduce prime fuzzy ideals over a noncommutative ring. This notion of primeness is equivalent to level cuts being crisp prime ideals. It also generalizes the one provided by Kumbhojkar and Bapat in [Not-so-fuzzy fuzzy…
Let $R$ be a commutative $G$-graded ring with a nonzero unity. In this article, we introduce the concept of graded radically principal ideals. A graded ideal $I$ of $R$ is said to be graded radically principal if $Grad(I)=Grad(\langle…
We study the two dual notions of prime avoidance and prime absorbance. We generalize the classical prime avoidance lemma to radical ideals. A number of new criteria are provided for an abstract ring to be C.P. (every set of primes satisfies…
The well-known Leibniz theorem (Leibniz Criterion or alternating series test) of convergence of alternating series is generalized for the case when the absolute value of terms of series are "not absolutely monotonously" convergent to zero.…
To each finitely aligned higher-rank graph $\Lambda$ and each $\mathbb{T}$-valued 2-cocycle on $\Lambda$, we associate a family of twisted relative Cuntz-Krieger algebras. We show that each of these algebras carries a gauge action, and…
We find necessary and sufficient conditions for the finite separability of finitely generated commutative rings. Namely, we prove that every such ring is a finite extension of its torsion ideal $I_k$ where $k$ is square-free, and $I_k$ is a…
We give simple necessary and sufficient conditions for the inclusion-exclusion identity to hold for an infinite countable number of sets. In terms of a random variable, whose range are nonnegative integers, this condition is equivalent to…
We prove a $p$-adic analog of Kunz's theorem: a $p$-adically complete noetherian ring is regular exactly when it admits a faithfully flat map to a perfectoid ring. This result is deduced from a more precise statement on detecting finiteness…
In this paper we provide necessary and sufficient conditions for strongly group graded rings to be simple. For a strongly group graded ring $R = \bigoplus_{g\in G} R_g$ the grading group $G$ acts, in a natural way, as automorphisms of the…
Topological quivers generalize the notion of directed graphs in which the sets of vertices and edges are locally compact (second countable) Hausdorff spaces. Associated to a topological quiver $Q$ is a $C^*$-correspondence, and in turn, a…
Given a contractive tuple of Hilbert space operators satisfying certain $A$-relations we show that there exists a unique minimal dilation to generators of Cuntz-Krieger algebras or its extension by compact operators. This Cuntz-Krieger…
We associate reduced and full C*-algebras to arbitrary rings and study the inner structure of these ring C*-algebras. As a result, we obtain conditions for them to be purely infinite and simple. We also discuss several examples.…
In this paper, we deal with the problem of uniqueness of minimal system of binomial generators of a semigroup ideal. Concretely, we give different necessary and/or sufficient conditions for uniqueness of such minimal system of generators.…
The existence of a maximal ideal in a general nontrivial commutative ring is tied together with the axiom of choice. Following Berardi, Valentini and thus Krivine but using the relative interpretation of negation (that is, as "implies 0 =…