Corollary

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

Algebra, chapter II, § 5. Extension of the ring of scalars

Let E, E' be two left A-modules; for every A-linear mapping $u : E \to E'$ , $v = l_B \otimes u$ is the unique B-linear mapping which renders commutative the diagram

\begin{array}{ccc} E & \xrightarrow{\phi_E} & E_{(B)} \\ | & & | \\ u \downarrow & & v \downarrow \\ E' & \xrightarrow{\phi_{E'}} & E'_{(B)} \end{array}

where $\phi_E$ and $\phi_{E'}$ are the canonical mappings.

It suffices to apply Proposition 1 to the A-homomorphism $\phi_{E'} \circ u : E \to E'_{(B)}$ .

The mapping $v$ defined in the above corollary is denoted by $\rho^*(u)$ or $u_{(B)}$ .

If $E''$ is a third left A-module and $v : E' \to E''$ an A-linear mapping, it is immediate that

(v \circ u)_{(B)} = v_{(B)} \circ u_{(B)}.

Extending the ring of operators of a module is a transitive operation; to be precise:

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

Cited by

Nothing.

Cites

Proposition 1