Bourbaki

Remark

Tag 05TO, permanent. Cite it as [Bourbaki, Tag 05TO].

Algebra, chapter II, § 7. Vector spaces

Let α^{-1}φ(x), β^{-1}φ(y) be two elements of E_{(K)}, with α ∈ A, β ∈ A, x ∈ E, y ∈ E, αβ ≠ 0. In order that α^{-1}φ(x) = β^{-1}φ(y), it is necessary and sufficient that βx − αy be a torsion element of E, for this relation is equivalent to βφ(x) = αφ(y), which may also be written φ(βx − αy) = 0.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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Cites

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Transcribed from the printed edition, 1998, Springer. Not translated.