Theorems · Theorem · linear algebra
Module.finrank_pos_iff
∀ {R : Type u_1} {M : Type u_2} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] [StrongRankCondition R]
[Module.Finite R M] [IsDomain R] [Module.IsTorsionFree R M], 0 < Module.finrank R M ↔ Nontrivial MAn R-finite torsion-free module has positive finrank iff it is nontrivial.
- Defined in
- Mathlib.LinearAlgebra.Dimension.Finite
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 101 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
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
- AddCommGroupstatement and proof · cited by 12,871
- Ringstatement and proof · cited by 7,463
- Cardinalproof · cited by 2,598
- Nontrivialstatement · cited by 2,416
- IsDomainstatement and proof · cited by 2,196
- Nat.cast_zeroproof · cited by 1,870
- Module.finrankstatement and proof · cited by 1,770
- Module.Finitestatement and proof · cited by 1,032
- Module.IsTorsionFreestatement and proof · cited by 600
- StrongRankConditionstatement and proof · cited by 286
- Module.finrank_eq_rankproof · cited by 29
Cited by8
Results whose statement or proof uses this declaration.
- Module.finrank_posproof · cited by 62
- finrank_span_singletonproof · cited by 16
- MeasureTheory.Measure.integral_comp_smulproof · cited by 3
- PrimeSpectrum.rankAtStalk_pos_iff_mem_range_comapproof · cited by 2
- Subalgebra.isSimpleOrder_of_finrankproof · cited by 1
- Module.End.iSup_maxGenEigenspace_eq_topproof · cited by 1
- Algebra.discr_powerBasis_eq_prod''proof · cited by 1
- LieAlgebra.hasTrivialRadical_of_isIrreducible_of_isFaithfulproof · cited by 0