Theorems · Theorem · commutative algebra
Submodule.IsPrincipal.generator_mem
∀ {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], Submodule.IsPrincipal.generator S ∈ S- Defined in
- Mathlib.RingTheory.PrincipalIdealDomain
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 21 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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.spanproof · cited by 1,504
- Submodule.subset_spanproof · cited by 234
- Set.mem_singletonproof · cited by 183
- Submodule.IsPrincipalstatement and proof · cited by 129
- Submodule.IsPrincipal.generatorstatement and proof · cited by 56
- Submodule.IsPrincipal.span_singleton_generatorproof · cited by 10
Cited by9
Results whose statement or proof uses this declaration.
- Submodule.IsPrincipal.prime_generator_of_isPrimeproof · cited by 4
- Submodule.basis_of_pid_auxproof · cited by 2
- Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRingproof · cited by 1
- Polynomial.annIdealGenerator_memproof · cited by 1
- UniqueFactorizationMonoid.of_forall_isPrincipal_of_height_eq_oneproof · cited by 1
- generator_maximal_submoduleImage_dvdproof · cited by 1
- IsPrime.to_maximal_idealproof · cited by 0
- Submodule.IsPrincipal.dvd_generator_span_iffproof · cited by 0
- Ideal.squarefree_span_singletonproof · cited by 0