Theorems · Theorem · linear algebra

Module.IsTorsion.rank_eq_zero

∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [Nontrivial R],
  Module.IsTorsion R M → Module.rank R M = 0

A torsion module has rank zero.

Defined in: Mathlib.LinearAlgebra.Dimension.Torsion.Finite
Cited by: 1 results in Mathlib
Foundations: Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Semiring AddCommMonoid Module Nontrivial

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Cites14

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coeproof · cited by 62,936
Modulestatement and proof · cited by 20,661
RingHom.idproof · cited by 18,349
Semiringstatement and proof · cited by 13,802
AddCommMonoidstatement and proof · cited by 12,281
LinearMapproof · cited by 10,215
mul_oneproof · cited by 3,885
Cardinalstatement · cited by 2,598
Nontrivialstatement and proof · cited by 2,416
nonZeroDivisorsproof · cited by 895
Module.rankstatement and proof · cited by 496
Module.IsTorsionstatement and proof · cited by 23

Cited by1

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Module.IsTorsion.finrank_eq_zeroproof · cited by 0