Theorem 2 (Krull–Remak–Schmidt)

Let A be a ring and M an A-module of finite length.

a) There exists a finite family $(M_i)_{i\in I}$ of indecomposable submodules of M such that $M =\bigoplus_{i\in I}M_i$, and the module M is semiprimordial.

b) Let $(M_i)_{i\in I}$ and $(M'_j)_{j\in J}$ be two finite families of indecomposable submodules of M such that $M =\bigoplus_{i\in I}M_i=\bigoplus_{j\in J}M'_j$. There exist a bijection $\sigma$ from I to J and an automorphism $u$ of M such that we have $u(M_i) = M'_{\sigma(i)}$ for every $i\in I$.

c) Let N be a direct factor submodule of M, and let $(M_i)_{i\in I}$ be a finite family of indecomposable submodules of M with direct sum M. There exists a subset J of I such that $\bigoplus_{i\in I-J}M_i$ is supplementary to N. The module N is isomorphic to $\bigoplus_{j\in J}M_j$.

d) Let N be an A-module. If there exists an integer $d >0$ such that the modules $M^d$ and $N^d$ are isomorphic, then the modules M and N are isomorphic.

e) Let N and P be A-modules of finite length. If the modules $M\oplus P$ and $N\oplus P$ are isomorphic, then M and N are isomorphic.

A module of finite length is both Artinian and Noetherian (VIII, p. 2, Proposition 1). Moreover, for a module of finite length, being indecomposable or primordial amounts to the same (VIII, p. 31, Proposition 4). Assertion a) then follows from Proposition 3 of VIII, p. 30. Assertions b), c), and e) follow from, respectively, Corollaries 2, 5, and 3 of Theorem 1 of VIII, p. 32. Finally, assertion d) follows from Corollary 4 of VIII, p. 35 because under the assumptions of d), the module N has finite length and is therefore semiprimordial by a).