Please use this identifier to cite or link to this item: https://hdl.handle.net/11147/10227
Title: Rings with few units and the infinitude of primes
Authors: Özcan, Hikmet Burak
Taşkın, Sedef
Keywords: Cardinality
Jacobson radical
Prime numbers
Issue Date: 2020
Publisher: Hacettepe Üniversitesi
Abstract: 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 R 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 R 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. © 2020, Hacettepe University. All rights reserved.
URI: https://doi.org/10.15672/hujms.649706
https://hdl.handle.net/11147/10227
https://search.trdizin.gov.tr/yayin/detay/499418
ISSN: 2651-477X
Appears in Collections:Mathematics / Matematik
Scopus İndeksli Yayınlar Koleksiyonu / Scopus Indexed Publications Collection
TR Dizin İndeksli Yayınlar / TR Dizin Indexed Publications Collection
WoS İndeksli Yayınlar Koleksiyonu / WoS Indexed Publications Collection

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