Proposition 2

Tag 00MJ, permanent. Cite it as [Bourbaki, Tag 00MJ].

Algebra, VIII, p. 378, § 20. Linear Representations of Algebras

Let E be a left A-module.

a) The set of coefficients of E is the union of the images of the A-linear mappings from E to $A^*$ .

b) Suppose, moreover, that E has finite dimension $n$ over K. Then the left A-module $\Theta_E(A)$ is isomorphic to a quotient of $E^n$ , the A-module E is isomorphic to a submodule of $\Theta_E(A)^n$ , and every element of $\Theta_E(A)$ is a coefficient of $E^n$ .

Assertion a) follows from the surjectivity of $c''_E$ .

Let us prove b). Let $(e^*_1, . . . , e^*_n)$ be a basis of $E^*$ over K. Since $\Theta_E(A)$ is generated by the coefficients of E, the A-linear mapping

(x_1, . . . , x_n)\mapsto \sum_{i=1}^nc_E(x_i, e^*_i)

from $E^n$ to $\Theta_E(A)$ is surjective. By a), every element of $\Theta_E(A)$ is a coefficient of $E^n$ . Moreover, the A-linear mapping

x\mapsto (c_E(x, e^*_1), . . . , c_E(x, e^*_n))

from E to $\Theta_E(A)^n$ is injective; this gives b).

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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