Theorems · Theorem · commutative algebra
RingHom.rangeS_top_of_surjective
∀ {R : Type u} {S : Type v} [inst : NonAssocSemiring R] [inst_1 : NonAssocSemiring S] (f : R →+* S),
Function.Surjective ⇑f → f.rangeS = ⊤The range of a surjective ring homomorphism is the whole of the codomain.
- Defined in
- Mathlib.Algebra.Ring.Subsemiring.Basic
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 25 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
- RingHomstatement and proof · cited by 10,189
- Top.topstatement · cited by 9,680
- NonAssocSemiringstatement and proof · cited by 805
- Subsemiringstatement · cited by 456
- RingHom.rangeSstatement · cited by 47
- RingHom.rangeS_top_iff_surjectiveproof · cited by 1
Cited by5
Results whose statement or proof uses this declaration.
- Subsemiring.range_fstproof · cited by 0
- Subsemiring.range_sndproof · cited by 0
- Subsemiring.map_comap_eq_self_of_surjectiveproof · cited by 0
- Subring.range_fstproof · cited by 0
- Subring.range_sndproof · cited by 0