Please can someone tell me the history of the simple argument that any maximal ideal of a commutative ring or distributive lattice is prime? (It is understood that we have found the maximal one using Zorn's Lemma.)
How were prime ideals found before the enactment of the Axiom of Choice? How do constructive algebraists find them now? What similar arguments (directly referring to polynomials or algebraic numbers) were used before ideals were invented?
My own interest is really in distributive lattices and locales, from a constructive point of view, but I am aware that these notions appeared much earlier for commutative rings.
Incorporating my comments:
For locales/frames, (the relevant analogue of) prime ideals are (formal) points. To be precise, these are completely coprime filters.
In the draft paper [http://www.paultaylor.eu/ASD/loccbv] on which I am working (and a propos of which I asked this question) I have a partly constructive argument that I also use to find points of inhabited overt subspaces.
I am really more interested in the history of the arguments than the definitions (or even concepts). Maybe factorisation is the relevant idea to pursue in order to answer my history question. As Mamuka says, it all comes from prime numbers and therefore probably from the argument by infinite descent in Elements VII 31.