Difference Rings

August 25, 2024
By Aathreya Kadambi

I wish we didn’t have to do all our proofs again and again for differences and differentials. I’m not sick of it (to be honest these proofs are pretty enjoyable), but everything just seems so similar, I would be surprised if there wasn’t a more generic algebraic object that emcompasses both differences and differentials. So after a consultation with ChatGPT, I’ve decided to read more about difference algebras, differential algebras, and “functional-theoretic algebra”.

I’m still waiting for people to board so I can get on my flight, so in the meantime, I’ll start this quick post on difference algebras. I may be quoting Wikipedia here, but the central idea behind difference algebras is “difference rings”. These are commutative rings with an (injective) endomorphism $\sigma$. Morphisms between difference rings are morphisms of rings taht commute with $\sigma$.