Theorems · Theorem · commutative algebra

Basis.mem_ideal_iff

∀ {ι : Type u_1} {R : Type u_2} {S : Type u_3} [inst : CommSemiring R] [inst_1 : Semiring S] [inst_2 : Algebra R S]
  {I : Ideal S} (b : Module.Basis ι R ↥I) {x : S}, x ∈ I ↔ ∃ c, x = c.sum fun i x => x • ↑(b i)

If I : Ideal S has a basis over R, x ∈ I iff it is a linear combination of basis vectors.

Defined in

Mathlib.RingTheory.Ideal.Basis

Cited by

0 results in Mathlib

Foundations

Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringSemiringAlgebra