This leads me to ask: is there a specific name for a morphism φ:A→B of commutative rings (or maybe just an inclusion) such that 𝔪 ↦ φ^−1(𝔪) defines a bijection between maximal ideals of B and those of A (i.e., Specmax(B) → Specmax(A))?
https://twitter.com/gro_tsen/status/1482307848637976579
(This is the case of the inclusion of the ring of smooth real functions inside the ring of continuous ones in the context of the tweet cited above.)
There are a number of sub-questions or variants, here:
‣ When does (pullback by) φ take a maximal ideal to a maximal ideal?
‣ When does it define a bijection on the prime ideals?
‣ Does the conjunction of the two above imply that it defines a bijection on the maximal ideals?
‣ When does (pullback by) φ take a maximal ideal to a maximal ideal?
‣ When does it define a bijection on the prime ideals?
‣ Does the conjunction of the two above imply that it defines a bijection on the maximal ideals?
‣ What if we replace “bijection” by “homeomorphism for the Zariski topology”?
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