Theorems · Theorem · commutative algebra

eq_bot_of_generator_maximal_map_eq_zero

∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {ι : Type u_1}
  (b : Module.Basis ι R M) {N : Submodule R M} {ϕ : M →ₗ[R] R},
  (∀ (ψ : M →ₗ[R] R), ¬Submodule.map ϕ N < Submodule.map ψ N) →
    ∀ [inst_3 : (Submodule.map ϕ N).IsPrincipal], Submodule.IsPrincipal.generator (Submodule.map ϕ N) = 0 → N = ⊥

Defined in

Mathlib.LinearAlgebra.FreeModule.PID

Cited by

0 results in Mathlib

Foundations

Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleSubmodule.IsPrincipal