### Abstract

We investigate properties of coset topologies on commutative domains with an identity, in particular, the S-coprime topologies de ned by Marko and Poru-bsky (2012) and akin to the topology de ned by Furstenberg (1955) in his proof of the in nitude of rational primes. We extend results about the in nitude of prime or maximal ideals related to the Dirichlet theorem on the in nitude of primes from Knopfmacher and Porubsky (1997), and correct some results from that paper. Then we determine cluster points for the set of primes and sets of primes appearing in arithmetic progressions in S-coprime topologies on ℤ. Finally, we give a new proof for the infinitude of prime ideals in number fields.

Original language | English (US) |
---|---|

Pages (from-to) | 221-237 |

Number of pages | 17 |

Journal | Colloquium Mathematicum |

Volume | 140 |

Issue number | 2 |

DOIs | |

State | Published - Jul 15 2015 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Colloquium Mathematicum*,

*140*(2), 221-237. https://doi.org/10.4064/cm140-2-5

}

*Colloquium Mathematicum*, vol. 140, no. 2, pp. 221-237. https://doi.org/10.4064/cm140-2-5

**Topological aspects of infinitude of primes in arithmetic progressions.** / Marko, František; Porubský, štefan.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Topological aspects of infinitude of primes in arithmetic progressions

AU - Marko, František

AU - Porubský, štefan

PY - 2015/7/15

Y1 - 2015/7/15

N2 - We investigate properties of coset topologies on commutative domains with an identity, in particular, the S-coprime topologies de ned by Marko and Poru-bsky (2012) and akin to the topology de ned by Furstenberg (1955) in his proof of the in nitude of rational primes. We extend results about the in nitude of prime or maximal ideals related to the Dirichlet theorem on the in nitude of primes from Knopfmacher and Porubsky (1997), and correct some results from that paper. Then we determine cluster points for the set of primes and sets of primes appearing in arithmetic progressions in S-coprime topologies on ℤ. Finally, we give a new proof for the infinitude of prime ideals in number fields.

AB - We investigate properties of coset topologies on commutative domains with an identity, in particular, the S-coprime topologies de ned by Marko and Poru-bsky (2012) and akin to the topology de ned by Furstenberg (1955) in his proof of the in nitude of rational primes. We extend results about the in nitude of prime or maximal ideals related to the Dirichlet theorem on the in nitude of primes from Knopfmacher and Porubsky (1997), and correct some results from that paper. Then we determine cluster points for the set of primes and sets of primes appearing in arithmetic progressions in S-coprime topologies on ℤ. Finally, we give a new proof for the infinitude of prime ideals in number fields.

UR - http://www.scopus.com/inward/record.url?scp=84982711366&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84982711366&partnerID=8YFLogxK

U2 - 10.4064/cm140-2-5

DO - 10.4064/cm140-2-5

M3 - Article

AN - SCOPUS:84982711366

VL - 140

SP - 221

EP - 237

JO - Colloquium Mathematicum

JF - Colloquium Mathematicum

SN - 0010-1354

IS - 2

ER -