Remark 2
Tag 00QF, permanent. Cite it as [Bourbaki, Tag 00QF].
Algebra, VIII, p. 455, Appendix 2. Determinants over a Noncommutative Field
The results of Subsections 1 through 4 generalize to the case when D is a local ring (VIII, p. 459, Exercise 2).
Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.
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Nothing.