Topological aspects of infinitude of primes in arithmetic progressions

František Marko, and Štefan Porubský. "Topological aspects of infinitude of primes in arithmetic progressions." Colloquium Mathematicae 140.2 (2015): 221-237. <http://eudml.org/doc/286554>.

@article{FrantišekMarko2015,
abstract = {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.},
author = {František Marko, Štefan Porubský},
journal = {Colloquium Mathematicae},
keywords = {coset topology; topological semigroup; topological density; Dirichlet theorem on primes; arithmetic progression; maximal ideal; ring of finite character; residually finite ring; infinitude of primes; pseudoprime},
language = {eng},
number = {2},
pages = {221-237},
title = {Topological aspects of infinitude of primes in arithmetic progressions},
url = {http://eudml.org/doc/286554},
volume = {140},
year = {2015},
}

TY - JOUR
AU - František Marko
AU - Štefan Porubský
TI - Topological aspects of infinitude of primes in arithmetic progressions
JO - Colloquium Mathematicae
PY - 2015
VL - 140
IS - 2
SP - 221
EP - 237
AB - 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.
LA - eng
KW - coset topology; topological semigroup; topological density; Dirichlet theorem on primes; arithmetic progression; maximal ideal; ring of finite character; residually finite ring; infinitude of primes; pseudoprime
UR - http://eudml.org/doc/286554
ER -