Proposition 5

(i) Every module admitting a finite presentation is finitely generated. (ii) If A is a left noetherian ring, every finitely generated a-module admits a finite presentation. (iii) Every finitely generated projective module admits a finite presentation.

Assertion (i) follows trivially from the definitions. Suppose A is left noetherian and E is finitely generated. There then exists a surjective homomorphism $u : L_0 \to E$, where $L_0$ is a free a-module having a finite basis; the kernel R of $u$ is finitely generated, hence there is a surjective homomorphism $v : L_1 \to R$ where $L_1$ is free with finite basis, and the exact sequence $L_1 \xrightarrow{v} L_0 \xrightarrow{u} E \to 0$ is a finite presentation of E; whence (ii).

Finally, suppose that E is a finitely generated projective module; it is then a direct factor of a finitely generated free module L_0 (II, p. 40, cor. 1); the kernel R of the surjective homomorphism L_0 → E is then isomorphic to a quotient of L_0, hence is finitely generated, and we conclude as above.