Topological Aspects of Infinitude of Primes in Arithmetic Progressions

0446412 - ÚI 2016 RIV PL eng J - Journal Article
Marko, F. - Porubský, Štefan
Topological Aspects of Infinitude of Primes in Arithmetic Progressions.
Colloquium Mathematicum. Roč. 140, č. 2 (2015), s. 221-237. ISSN 0010-1354. E-ISSN 1730-6302
R&D Projects: GA ČR(CZ) GAP201/12/2351
Institutional support: RVO:67985807
Keywords : coset topology * topological semigroup * topological density * Dirichlet theorem on primes * arithmetical progression * maximal ideal * ring of finite character * residually finite ring * infinitude of primes * pseudoprime
Subject RIV: BA - General Mathematics
Impact factor: 0.333, year: 2015

We investigate properties of coset topologies on commutative domains with an identity, in particular, the S-coprime topologies defined by Marko and Porubský (2012) and akin to the topology defined by Furstenberg (1955) in his proof of the infinitude of rational primes. We extend results about the infinitude of prime or maximal ideals related to the Dirichlet theorem on the infinitude of primes from Knopfmacher and Porubský (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.
Permanent Link: http://hdl.handle.net/11104/0248415