Corollary

Tag 008D, permanent. Cite it as [Bourbaki, Tag 008D].

Algebra, VIII, p. 123, § 7. Simple Rings

Let $k$ be a commutative field, and let A be a simple $k$ -algebra of finite degree over $k$ . Every simple A-module is finite-dimensional over $k$ . Two A-modules are isomorphic if and only if their dimensions over $k$ are equal.

Every simple A-module is isomorphic to a quotient of $A_s$ , hence finite-dimensional over $k$ . The corollary then follows from Proposition 2, c).

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.

Cited by

Nothing.

Cites

Proposition 2