## 7 Citations

### Radicals of principal ideals and the class group of a Dedekind domain

- MathematicsPacific Journal of Mathematics
- 2021

For a Dedekind domain $D$, let $\mathcal{P}(D)$ be the set of ideals of $D$ that are radical of a principal ideal. We show that, if $D,D'$ are Dedekind domains and there is an order isomorphism…

### The Golomb space is topologically rigid

- Mathematics
- 2019

The $Golomb$ $space$ $\mathbb N_\tau$ is the set $\mathbb N$ of positive integers endowed with the topology $\tau$ generated by the base consisting of arithmetic progressions $\{a+bn:n\ge 0\}$ with…

### Homeomorphisms of the space of non-zero integers with the Kirch topology

- Mathematics
- 2020

Abstract. The Golomb (resp. Kirch) topology on the set Z of nonzero integers is generated by the base consisting of arithmetic progressions a+ bZ = {a+ bn : n ∈ Z} where a ∈ Z and b is a…

### The Golomb Topology of Polynomial Rings

- MathematicsQuaestiones Mathematicae
- 2020

Abstract We study properties of the Golomb topology on polynomial rings over fields, in particular trying to determine conditions under which two such spaces are not homeomorphic. We show that if K…

### The Golomb topology of polynomial rings, II

- Mathematics
- 2022

. We study the interplay of the Golomb topology and the algebraic structure in polynomial rings K [ X ] over a ﬁeld K . In particular, we focus on inﬁnite ﬁelds K of positive characteristic such that…

## References

SHOWING 1-10 OF 22 REFERENCES

### On continuous self-maps and homeomorphisms of the Golomb space

- MathematicsCommentationes Mathematicae Universitatis Carolinae
- 2019

The Golomb space $\mathbb N_\tau$ is the set $\mathbb N$ of positive integers endowed with the topology $\tau$ generated by the base consisting of arithmetic progressions $\{a+bn\}_{n=0}^\infty$ with…

### On the Infinitude of Primes

- Mathematics
- 1955

In this note we would like to offer an elementary “topological” proof of the infinitude of the prime numbers. We introduce a topology into the space of integers S, by using the arithmetic…

### COUNTABLE METRIC SPACES WITHOUT ISOLATED POINTS

- Mathematics
- 2005

The theorem is remarkable, and gives some apparently counter-intuitive examples of spaces homeomorphic to the usual Q. Consider the “Sorgenfrey topology on Q,” which has the collection {(p, q] : p, q…

### A note on Golomb topologies

- Mathematics
- 2019

Abstract In 1959 Golomb defined a connected topology on ℤ. An analogous Golomb topology on an arbitrary integral domain was defined first by Knopfmacher-Porubský [KP97] and then again in a recent…

### The Euclidean Criterion for Irreducibles

- MathematicsAm. Math. Mon.
- 2017

Euclid's proof of the infinitude of prime numbers is recast as a Euclidean criterion for a domain to have infinitely many atoms and it is shown that this criterion applies even in certain domains in which not all nonzero nonunits factor into products of irreducibles.

### Introduction to analytic number theory

- Mathematics
- 1976

This is the first volume of a two-volume textbook which evolved from a course (Mathematics 160) offered at the California Institute of Technology during the last 25 years. It provides an introduction…

### Algebraic Number Theory

- Mathematics
- 1999

I: Algebraic Integers.- II: The Theory of Valuations.- III: Riemann-Roch Theory.- IV: Abstract Class Field Theory.- V: Local Class Field Theory.- VI: Global Class Field Theory.- VII: Zeta Functions…

### Introduction to commutative algebra

- Mathematics
- 1969

* Introduction * Rings and Ideals * Modules * Rings and Modules of Fractions * Primary Decomposition * Integral Dependence and Valuations * Chain Conditions * Noetherian Rings * Artin Rings *…