Theorems · Theorem · commutative algebra
isReduced_of_injective
∀ {R : Type u_1} {S : Type u_2} [inst : MonoidWithZero R] [inst_1 : MonoidWithZero S] {F : Type u_3}
[inst_2 : FunLike F R S] [MonoidWithZeroHomClass F R S] (f : F), Function.Injective ⇑f → ∀ [IsReduced S], IsReduced R- Defined in
- Mathlib.RingTheory.Nilpotent.Defs
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 13 from the axioms · uses Classical.choice
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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
- FunLikestatement and proof · cited by 2,560
- map_zeroproof · cited by 1,614
- MonoidWithZerostatement and proof · cited by 456
- IsNilpotentproof · cited by 248
- IsReducedstatement and proof · cited by 98
- MonoidWithZeroHomClassstatement and proof · cited by 37
- IsNilpotent.eq_zeroproof · cited by 13
- IsNilpotent.mapproof · cited by 12
Cited by7
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.isReduced_of_isOpenImmersionproof · cited by 5
- AlgebraicGeometry.IsReduced.of_openCoverproof · cited by 2
- AlgebraicGeometry.IsSchemeTheoreticallyDominant.isReducedproof · cited by 1
- Algebra.isGeometricallyReduced_field_iffproof · cited by 1
- Algebra.isReduced_of_isGeometricallyReducedproof · cited by 0
- AlgebraicGeometry.affine_isReduced_iffproof · cited by 0
- Algebra.IsGeometricallyReduced.of_injectiveproof · cited by 0