Theorems · Theorem · commutative algebra

Ideal.comap_map_eq_self_of_isMaximal

∀ {R : Type u} {S : Type v} [inst : CommSemiring R] [inst_1 : CommSemiring S] (f : R →+* S) {p : Ideal R}
  [hP' : p.IsMaximal], Ideal.map f p ≠ ⊤ → Ideal.comap f (Ideal.map f p) = p

For a maximal ideal p of R, p extended to S and restricted back to R is p if its image in S is not equal to ⊤.

Defined in

Mathlib.RingTheory.Ideal.Maps

Cited by

2 results in Mathlib

Foundations

Depth 29 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringCommSemiringIdeal.IsMaximal