Corollary 1

Let A be a ring, and let M be an A-module.

a) If the module M is Noetherian, then every surjective endomorphism of M is bijective.

b) If the module M is Artinian, then every injective endomorphism of M is bijective.

c) If the module M has finite length, then every injective or surjective endomorphism of M is bijective.

d) If the ring A is commutative and the A-module M is finitely generated, then every surjective endomorphism of M is bijective.

Let $u$ be an endomorphism of the A-module M. We use the notation introduced at the beginning of this subsection. If the endomorphism $u$ is surjective, then we have $I_{\infty}= M$. Assertion a) then follows from Proposition 2, a) and Remark 3. Likewise, if the endomorphism $u$ is injective, we have $N_{\infty}= 0$. Assertion b) therefore follows from Proposition 2, b) and Remark 3. Assertion c) follows immediately from a) and b).

Now, suppose that the ring A is commutative, the A-module M is finitely generated, and the endomorphism $u$ is surjective. Let us prove that $u$ is injective. Let $x$ be an element of M such that $u(x) = 0$. Choose a finite generating family $(x_i)_{i\in I}$ of the A-module M and, for every $i\in I$, an element $y_i$ of M such that $u(y_i) =x_i$. There exist families $(a_i)_{i\in I}, (b_{ij})_{(i,j)\in I\times I}$, and $(c_{ij})_{(i,j)\in I\times I}$ of elements of A such that we have

for every $j\in I$. Let $A'$ be a Noetherian subring of A containing the elements $a_i,b_{ij}$, and $c_{ij}$ (VIII, p. 12, Corollary 3). Let $M'$ be the $A'$-submodule of M generated by the family $(x_i)_{i\in I}$. We have $u(x_j)\in M',y_j\in M'$, and $u(y_j) =x_j$ for every $j\in I$; hence $u$ defines, by restriction, a surjective endomorphism $u'$ of the $A'$-module $M'$. Since the ring $A'$ is Noetherian, the finitely generated $A'$-module $M'$ is Noetherian (VIII, p. 7, Proposition 4 a)). By a), the endomorphism $u'$ of $M'$ is bijective. By construction, $x$ belongs to $M'$, and we have $u'(x) =u(x) = 0$. We therefore have $x= 0$, which proves d).