Theorems · Theorem · commutative algebra
Submodule.IsPrincipal.mem_iff_eq_smul_generator
∀ {R : Type u} {M : Type v} [inst : AddCommMonoid M] [inst_1 : Semiring R] [inst_2 : Module R M] (S : Submodule R M)
[inst_3 : S.IsPrincipal] {x : M}, x ∈ S ↔ ∃ s, x = s • Submodule.IsPrincipal.generator S- Defined in
- Mathlib.RingTheory.PrincipalIdealDomain
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 28 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.
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Submodulestatement and proof · cited by 7,192
- Submodule.IsPrincipalstatement and proof · cited by 129
- Submodule.IsPrincipal.generatorstatement and proof · cited by 56
- Submodule.IsPrincipal.span_singleton_generatorproof · cited by 10
Cited by3
Results whose statement or proof uses this declaration.
- Submodule.IsPrincipal.mem_iff_generator_dvdproof · cited by 11
- Submodule.IsPrincipal.contentIdeal_generator_dvd_coeffproof · cited by 2
- Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRingproof · cited by 1