Theorems · Theorem · commutative algebra
dvd_generator_iff
∀ {R : Type u_1} [inst : CommSemiring R] {I : Ideal R} [inst_1 : Submodule.IsPrincipal I] {x : R},
x ∈ I → (x ∣ Submodule.IsPrincipal.generator I ↔ I = Ideal.span {x})- Defined in
- Mathlib.LinearAlgebra.FreeModule.PID
- Cited by
- 1 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.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- CommSemiringstatement and proof · cited by 10,911
- Submoduleproof · cited by 7,192
- Idealstatement and proof · cited by 4,748
- Ideal.spanstatement and proof · cited by 948
- Submodule.IsPrincipalstatement and proof · cited by 129
- Submodule.IsPrincipal.generatorstatement and proof · cited by 56
- Ideal.span_singleton_le_iff_memproof · cited by 29
- Submodule.span_singleton_le_iff_memproof · cited by 21
- Submodule.IsPrincipal.span_singleton_generatorproof · cited by 10
Cited by1
Results whose statement or proof uses this declaration.
- generator_maximal_submoduleImage_dvdproof · cited by 1