Theorems · Theorem · commutative algebra
Submodule.mem_colon_singleton
∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {N : Submodule R M}
{x : M} {r : R}, r ∈ N.colon {x} ↔ r • x ∈ N- Defined in
- Mathlib.RingTheory.Ideal.Colon
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 65 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringAddCommMonoidModule
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.
- Setstatement · cited by 53,352
- 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
- Idealstatement · cited by 4,748
- Submodule.colonstatement · cited by 80
Cited by8
Results whose statement or proof uses this declaration.
- Algebra.denominator_dvd_iffproof · cited by 3
- biUnion_associatedPrimes_eq_zero_divisorsproof · cited by 2
- exists_le_isAssociatedPrime_of_isNoetherianRingproof · cited by 2
- associatedPrimes.subset_union_of_exactproof · cited by 2
- IsAssociatedPrime.eq_radicalproof · cited by 1
- Submodule.exists_eq_colon_of_mem_minimalPrimesproof · cited by 1
- Submodule.IsPrimary.radical_colon_singleton_of_notMemproof · cited by 1