Rings with few units and the infinitude of primes
| Autor: | Hikmet Burak Özcan, Sedef Taşkin |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Statistics and Probability
Ring (mathematics) Matematik Algebra and Number Theory Mathematics::General Mathematics Prime number Jacobson radical Commutative ring prime number cardinality Jacobson radical Mathematical proof Combinatorics Cardinality Integer Domain (ring theory) Geometry and Topology Analysis Mathematics |
| Zdroj: | Volume: 49, Issue: 6 2071-2073 Hacettepe Journal of Mathematics and Statistics |
| ISSN: | 2651-477X |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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