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Year 2020, Volume: 49 Issue: 6, 2071 - 2073, 08.12.2020
https://doi.org/10.15672/hujms.649706

Abstract

References

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

Rings with few units and the infinitude of primes

Year 2020, Volume: 49 Issue: 6, 2071 - 2073, 08.12.2020
https://doi.org/10.15672/hujms.649706

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.

References

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

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Hikmet Burak Özcan This is me 0000-0002-8684-7311

Sedef Taşkın 0000-0002-5835-1642

Publication Date December 8, 2020
Published in Issue Year 2020 Volume: 49 Issue: 6

Cite

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. December 2020;49(6):2071-2073. doi:10.15672/hujms.649706
Chicago Özcan, Hikmet Burak, and Sedef Taşkın. “Rings With Few Units and the Infinitude of Primes”. Hacettepe Journal of Mathematics and Statistics 49, no. 6 (December 2020): 2071-73. https://doi.org/10.15672/hujms.649706.
EndNote Özcan HB, Taşkın S (December 1, 2020) Rings with few units and the infinitude of primes. Hacettepe Journal of Mathematics and Statistics 49 6 2071–2073.
IEEE H. B. Özcan and S. Taşkın, “Rings with few units and the infinitude of primes”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 6, pp. 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 (December 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 and Sedef Taşkın. “Rings With Few Units and the Infinitude of Primes”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 6, 2020, pp. 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.