Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article A Note on Points on Algebraic Sets(Hacettepe Üniversitesi, 2021) Çam Çelik, Şermin; Göral, HaydarIn this short note, we count the points on algebraic sets which lie in a subset of a domain. It is proved that the set of points on algebraic sets coming from certain subsets of a domain has the full asymptotic. This generalizes the first theorem of [E. Alkan and E.S. Yoruk, Statistics and characterization of matrices by determinant and trace, Ramanujan J., 2019] and also anwers some questions from the same article.Article Citation - WoS: 2Citation - Scopus: 2Rings With Few Units and the Infinitude of Primes(Hacettepe Üniversitesi, 2020) Özcan, Hikmet Burak; Taşkın, SedefIn 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.