Rings with few units and the infinitude of primes

Hikmet Burak ÖZCAN; Sedef TAŞKIN

doi:https://doi.org/10.15672/hujms.649706

Makale detayları

Rings with few units and the infinitude of primes

Yazarlar:

Hikmet Burak ÖZCAN | Sedef TAŞKIN

Yayıncı:

Hacettepe University

Ana Başlıklar:

Matematik

Dergi:

Hacettepe Journal of Mathematics and Statistics 2020 Vol.49 Issue 6, pp.2071-2073

Belge Tipi:

Makale

Yayın Tarihi:

12/2020

Belge Dili:

İngilizce

Document ID (Org No.):

2000450880

Sıklık:

İki Aylık

eISSN:

2651-477X

pISSN:

2651-477X

DOI:

https://doi.org/10.15672/hujms.649706

Özet:

 

In this short note, our aim is to provide novel proofs for the infinitude of primes in an algebraic way. It's thought that the first proof for the infinitude of primes was given by the Ancient Greek mathematician Euclid. To date, most of the proofs have been based on the fact that every positive integer greater than 1 can be written as a product of prime numbers. However, first we are going to prove a ring theoretic fact that if RRR is an infinite commutative ring with unity and the cardinality of the set of invertible elements is strictly less than the cardinality of the ring, then there are infinitely many maximal ideals. This fact leads to an elegant proof for the infinitude of primes. In addition, under the same cardinality assumption, we consider the special case in which RRR is a unique factorization domain (for short UFD) and establish another ring theoretic result. Thanks to it, we give a second proof of the infinitude of primes.

Makale detayları

Rings with few units and the infinitude of primes

Yazarlar: Hikmet Burak ÖZCAN | Sedef TAŞKIN

Yayıncı: Hacettepe University

Ana Başlıklar: Matematik

Dergi: Hacettepe Journal of Mathematics and Statistics 2020 Vol.49 Issue 6, pp.2071-2073

Belge Tipi: Makale

Yayın Tarihi: 12/2020

Belge Dili: İngilizce

Document ID (Org No.): 2000450880

Sıklık: İki Aylık

eISSN: 2651-477X

pISSN: 2651-477X

DOI: https://doi.org/10.15672/hujms.649706

Sayı

Yazarlar:

Hikmet Burak ÖZCAN | Sedef TAŞKIN

Yayıncı:

Hacettepe University

Ana Başlıklar:

Matematik

Dergi:

Hacettepe Journal of Mathematics and Statistics 2020 Vol.49 Issue 6, pp.2071-2073

Belge Tipi:

Makale

Yayın Tarihi:

12/2020

Belge Dili:

İngilizce

Document ID (Org No.):

2000450880

Sıklık:

İki Aylık

eISSN:

2651-477X

pISSN:

2651-477X

DOI:

https://doi.org/10.15672/hujms.649706