Theorems · Theorem · commutative algebra
Submodule.IsPrincipal.eq_bot_iff_generator_eq_zero
∀ {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], S = ⊥ ↔ Submodule.IsPrincipal.generator S = 0- Defined in
- Mathlib.RingTheory.PrincipalIdealDomain
- Cited by
- 7 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.
Cites9
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
- Bot.botstatement and proof · cited by 4,720
- Submodule.IsPrincipalstatement and proof · cited by 129
- Submodule.IsPrincipal.generatorstatement · cited by 56
- Submodule.IsPrincipal.span_singleton_generatorproof · cited by 10
- Submodule.span_singleton_eq_botproof · cited by 7
Cited by7
Results whose statement or proof uses this declaration.
- Submodule.IsPrincipal.prime_generator_of_isPrimeproof · cited by 4
- eq_bot_of_generator_maximal_submoduleImage_eq_zeroproof · cited by 1
- Module.Flat.flat_iff_torsion_eq_bot_of_isBezoutproof · cited by 1
- IsDiscreteValuationRing.ideal_eq_span_pow_irreducibleproof · cited by 1
- eq_bot_of_generator_maximal_map_eq_zeroproof · cited by 0
- IsPrime.to_maximal_idealproof · cited by 0
- Submodule.IsPrincipal.generator_botproof · cited by 0