In the previous seminar, Nitin Bartwal presented a topological proof of the infinitude of primes. Motivated by that beautiful talk, I will present an algebraic approach to the same classical problem. It is well known that the first proof of the infinitude of primes was given by Euclid, and since then many different proofs have been discovered. Most of them rely on the fundamental fact that every integer greater than 1 can be expressed as a product of primes.
The central idea of this talk is that in an infinite commutative ring with unity, if the group of units is “small” compared to the ring itself, then the ring must have infinitely many maximal ideals. This observation provides a direct and elegant algebraic proof of Euclid’s theorem. Moreover, when the ring is a unique factorization domain, the same principle yields yet another proof of the result.
The aim of the talk is to illustrate how a classical theorem can be rediscovered in new settings, where even a simple structural property of rings opens a pathway to infinity.
Although inspired by the last talk, this presentation will be self-contained. Only basic ring theory concepts will be assumed.
