Bourbaki

Corollary

Tag 060M, permanent. Cite it as [Bourbaki, Tag 060M].

Algebra, chapter III, § 4. Tensor products of algebras

If Z and Z' are the respective centres of E and F, the centre of E \otimesK F is Z \otimes_K Z'*.

Let E and F be two subalgebras of an algebra G over a commutative field K; suppose that every element of E commutes with every element of F. Then the canonical injections i:E→G,j:F→Gi : E \to G, j : F \to G define a canonical homomorphism h=i⊗j:E⊗KF→Gh = i \otimes j : E \otimes_K F \to G (no. 2, Proposition 5) such that

(i⊗j)(x⊗y)=xyfor x∈E,y∈F. (i \otimes j)(x \otimes y) = xy \quad \text{for } x \in E, y \in F.

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

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Cites

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