Remark

Proposition 5 applies, in particular, to the following two cases:

a) Let A be an algebra over a commutative field K, and let M be a faithful finite-dimensional module over K. Then M is an A-module of finite length, and the countermodule of M is finitely generated. The ring A is left Artinian (VIII, p. 9, Proposition 6). There exists a Jordan–Hölder series $(M_i)_{0\leqslant i\leqslant n}$ of the A-module M, and every simple A-module is isomorphic to one of the modules $M_i/M_{i+1}$ for $0\leqslant i\leqslant n-1$.

b) Let A be a left Artinian ring. The module $A_s$ has finite length (VIII, p. 6, Theorem 1). Since the A-module $A_s$ has finite length, there exists a decreasing sequence $(\mathfrak{a}_i)_{0\leqslant i\leqslant n}$ of left ideals of A such that $\mathfrak{a}_0= A$ and $\mathfrak{a}_n= 0$ and that the A-modules $S_i=\mathfrak{a}_{i-1}/\mathfrak{a}_i$ are simple for $1\leqslant i\leqslant n$. Then every simple A-module is isomorphic to one of the modules $S_1, . . . ,S_n$.