Theorems · Theorem · commutative algebra

Module.sum_smul_eq_zero_of_isTrivialRelation

∀ {R : Type u_1} {M : Type u_2} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {ι : Type u_3}
  [inst_3 : Fintype ι] {f : ι → R} {x : ι → M}, Module.IsTrivialRelation f x → ∑ i, f i • x i = 0

If the relation given by $(f_i)_{i \in \iota}$ and $(x_i)_{i \in \iota}$ is trivial, then $\sum_i f_i x_i$ is actually equal to $0$.

Defined in

Mathlib.RingTheory.Flat.EquationalCriterion

Cited by

0 results in Mathlib

Foundations

Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleFintype