Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2020, Cilt: 49 Sayı: 6, 2071 - 2073, 08.12.2020
https://doi.org/10.15672/hujms.649706

Öz

Kaynakça

  • [1] T.Y. Lam, A First Course in Noncommutative Rings, Springer, 1999.

Rings with few units and the infinitude of primes

Yıl 2020, Cilt: 49 Sayı: 6, 2071 - 2073, 08.12.2020
https://doi.org/10.15672/hujms.649706

Öz

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.

Kaynakça

  • [1] T.Y. Lam, A First Course in Noncommutative Rings, Springer, 1999.
Toplam 1 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Matematik
Yazarlar

Hikmet Burak Özcan Bu kişi benim 0000-0002-8684-7311

Sedef Taşkın 0000-0002-5835-1642

Yayımlanma Tarihi 8 Aralık 2020
Yayımlandığı Sayı Yıl 2020 Cilt: 49 Sayı: 6

Kaynak Göster

APA Özcan, H. B., & Taşkın, S. (2020). Rings with few units and the infinitude of primes. Hacettepe Journal of Mathematics and Statistics, 49(6), 2071-2073. https://doi.org/10.15672/hujms.649706
AMA Özcan HB, Taşkın S. Rings with few units and the infinitude of primes. Hacettepe Journal of Mathematics and Statistics. Aralık 2020;49(6):2071-2073. doi:10.15672/hujms.649706
Chicago Özcan, Hikmet Burak, ve Sedef Taşkın. “Rings With Few Units and the Infinitude of Primes”. Hacettepe Journal of Mathematics and Statistics 49, sy. 6 (Aralık 2020): 2071-73. https://doi.org/10.15672/hujms.649706.
EndNote Özcan HB, Taşkın S (01 Aralık 2020) Rings with few units and the infinitude of primes. Hacettepe Journal of Mathematics and Statistics 49 6 2071–2073.
IEEE H. B. Özcan ve S. Taşkın, “Rings with few units and the infinitude of primes”, Hacettepe Journal of Mathematics and Statistics, c. 49, sy. 6, ss. 2071–2073, 2020, doi: 10.15672/hujms.649706.
ISNAD Özcan, Hikmet Burak - Taşkın, Sedef. “Rings With Few Units and the Infinitude of Primes”. Hacettepe Journal of Mathematics and Statistics 49/6 (Aralık 2020), 2071-2073. https://doi.org/10.15672/hujms.649706.
JAMA Özcan HB, Taşkın S. Rings with few units and the infinitude of primes. Hacettepe Journal of Mathematics and Statistics. 2020;49:2071–2073.
MLA Özcan, Hikmet Burak ve Sedef Taşkın. “Rings With Few Units and the Infinitude of Primes”. Hacettepe Journal of Mathematics and Statistics, c. 49, sy. 6, 2020, ss. 2071-3, doi:10.15672/hujms.649706.
Vancouver Özcan HB, Taşkın S. Rings with few units and the infinitude of primes. Hacettepe Journal of Mathematics and Statistics. 2020;49(6):2071-3.