Theorems · Theorem · commutative algebra
Submodule.IsPrincipal.mem_iff_generator_dvd
∀ {R : Type u} [inst : CommSemiring R] (S : Ideal R) [inst_1 : Submodule.IsPrincipal S] {x : R},
x ∈ S ↔ Submodule.IsPrincipal.generator S ∣ x- Defined in
- Mathlib.RingTheory.PrincipalIdealDomain
- Cited by
- 11 results in Mathlib
- Foundations
- Depth 29 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommSemiringstatement and proof · cited by 10,911
- Idealstatement and proof · cited by 4,748
- mul_commproof · cited by 2,262
- Submodule.IsPrincipalstatement and proof · cited by 129
- Submodule.IsPrincipal.generatorstatement and proof · cited by 56
- Submodule.IsPrincipal.mem_iff_eq_smul_generatorproof · cited by 3
Cited by11
Results whose statement or proof uses this declaration.
- Nat.setGcd_dvd_of_memproof · cited by 5
- Submodule.IsPrincipal.prime_generator_of_isPrimeproof · cited by 4
- Algebra.denominator_dvd_iffproof · cited by 3
- IsBezout.gcd_dvd_leftproof · cited by 3
- IsBezout.gcd_dvd_rightproof · cited by 3
- Ideal.torsionOf_eq_span_pow_pOrderproof · cited by 2
- generator_maximal_submoduleImage_dvdproof · cited by 1
- Submodule.IsPrincipal.generator_submoduleImage_dvd_of_memproof · cited by 1
- IsPrime.to_maximal_idealproof · cited by 0
- Submodule.IsPrincipal.dvd_generator_span_iffproof · cited by 0
- Submodule.IsPrincipal.generator_map_dvd_of_memproof · cited by 0