Theorems · Theorem · commutative algebra
IsPrincipalIdealRing.of_surjective
∀ {R : Type u} {S : Type u_1} {F : Type u_3} [inst : Semiring R] [inst_1 : Semiring S] [inst_2 : FunLike F R S]
[RingHomClass F R S] [IsPrincipalIdealRing R] (f : F), Function.Surjective ⇑f → IsPrincipalIdealRing SThe surjective image of a principal ideal ring is again a principal ideal ring.
- Defined in
- Mathlib.RingTheory.PrincipalIdealDomain
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 30 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Semiringstatement and proof · cited by 13,802
- Idealproof · cited by 4,748
- FunLikestatement and proof · cited by 2,560
- RingHomClassstatement and proof · cited by 193
- IsPrincipalIdealRingstatement and proof · cited by 131
- Ideal.IsPrincipal.of_comapproof · cited by 1
Cited by4
Results whose statement or proof uses this declaration.
- Valuation.Integers.isPrincipalIdealRing_iff_not_denselyOrderedproof · cited by 1
- isPrincipalIdealRing_pi_iffproof · cited by 0
- isPrincipalIdealRing_prod_iffproof · cited by 0
- Rat.classNumber_eqproof · cited by 0