Strong rank condition for commutative rings

We prove that any nontrivial commutative ring satisfies StrongRankCondition, meaning that if there is an injective linear map (Fin n → R) →ₗ[R] Fin m → R, then n ≤ m. This implies that any commutative ring satisfies InvariantBasisNumber: the rank of a finitely generated free module is well defined.

Main result

• commRing_strongRankCondition R : R has the StrongRankCondition.

The commRing_strongRankCondition comes from CommRing.orzechProperty, proved in Mathlib/RingTheory/FiniteType.lean, which states that any commutative ring satisfies the OrzechProperty, that is, for any finitely generated R-module M, any surjective homomorphism f : N → M from a submodule N of M to M is injective.

References

• [Orzech, Morris. Onto endomorphisms are isomorphisms][orzech1971]
• [Djoković, D. Ž. Epimorphisms of modules which must be isomorphisms][djokovic1973]
• [Ribenboim, Paulo. Épimorphismes de modules qui sont nécessairement des isomorphismes][ribenboim1971]

@[instance 100]

instance commRing_strongRankCondition (R : Type u_1) [CommRing R] [Nontrivial R] : StrongRankCondition R

Any nontrivial commutative ring satisfies the StrongRankCondition.

Equations

• ⋯ = ⋯