Proposition 4

a) A primordial module is indecomposable.

b) An indecomposable module of finite length is primordial.

Let M be an A-module. Suppose that M is primordial; let $e$ be an idempotent in the local ring End$_A(M)$. Since $e^2=e$, either $e$ is invertible and we have $e= 1$, or $1-e$ is invertible and we have $e= 0$. This proves that M is indecomposable (VIII, p. 30).

Now, suppose that M is indecomposable and of finite length. By Proposition 2, c) of VIII, p. 27, every endomorphism of M is invertible or nilpotent; the ring End$_A(M)$ is therefore local by Example 2 of VIII, p. 26.

The $\mathbf{Z}$-module $\mathbf{Z}$ is indecomposable, Noetherian, but not Artinian. Its endomorphism ring is isomorphic to $\mathbf{Z}$, hence is not local. Consequently, $\mathbf{Z}$ is not a primordial $\mathbf{Z}$-module.

$*$Let $p$ be a prime number. The endomorphism ring of the $\mathbf{Z}$-module $\mathbf{Q}_p/\mathbf{Z}_p$ is isomorphic to the local ring $\mathbf{Z}_p$ (cf. VII, §4, p. 65, Exercise 13); it is therefore a primordial $\mathbf{Z}$-module.

An injective module is indecomposable if and only if it is primordial (X, §1, n$^o9$, p. 21, proposition $14$)$.*$