## ยง Why is the spectrum of a ring called so?

I've been watching Ravi Vakil's excellent "pseudolectures" on algebraic geometry,
aptly titled
AGITTC: Algebraic geometry in the time of Covid.
In lecture 3, there was a discussion going on in the sidebar chat where a user
said that the name "prime sprectrum" came from something to do with quantum
mechanics. To quote:
letheology: spectrum of light -> eigenvalues of the hamiltonian operator -> prime ideal of the polynomial ring of the operator

I don't know what the prime ideal of the polynomial ring of the operator is,
so let's find out! I got a somewhat incomplete answer on `math.se`

Another user said:
Lukas H: I like the definition of Spec A that doesn't include the word prime ideal, by a colimit of Hom(A, k) where k run over all fields and the maps are morphisms that make the diagrams commute.

That's a pretty crazy definition. One can apparently find this definition
in Peter Schloze's notes on AG. I got an answer for this on `math.se`