Theorems · Theorem · linear algebra
Submodule.IsPrincipal.principal
∀ {R : Type u_1} {M : Type u_4} {inst : Semiring R} {inst_1 : AddCommMonoid M} {inst_2 : Module R M} (S : Submodule R M)
[self : S.IsPrincipal], ∃ a, S = R ∙ a- Defined in
- Mathlib.LinearAlgebra.Span.Defs
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 18 from the axioms · uses propext, Quot.sound
- Assumes
- Submodule.IsPrincipal
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
- Submodule.spanstatement · cited by 1,504
- Submodule.IsPrincipalstatement and proof · cited by 129
Cited by11
Results whose statement or proof uses this declaration.
- Submodule.IsPrincipal.generatorproof · cited by 56
- Ideal.span_singleton_generatorproof · cited by 17
- Submodule.IsPrincipal.span_singleton_generatorproof · cited by 10
- IsDiscreteValuationRing.exists_irreducibleproof · cited by 8
- ClassGroup.mk_eq_one_iffproof · cited by 4
- IsBezout.TFAEproof · cited by 2
- IsDiscreteValuationRing.iff_pid_with_one_nonzero_primeproof · cited by 2
- Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRingproof · cited by 1
- Ideal.isPrime_iff_of_isPrincipalIdealRingproof · cited by 1
- IsGCDMonoid.isPrincipal_of_exists_mul_ne_zero_isPrincipalproof · cited by 1
- Module.IsPrincipal.of_surjectiveproof · cited by 1