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Choose $s\ne 0$ in the simple module $S$. The map $\varphi :A\to S$, $a\mapsto as$, is an $A$-module homomorphism whose image is the nonzero submodule $As$, hence all of $S$. Thus the first isomorphism theorem for modules givesThe kernel is a maximal left ideal because the quotient is simple. Whenever the regular module has finite length, as it does for a finite-dimensional algebra, take a composition series of the kernel and append $\ker \varphi <A$. Its last factor is $S$, so $S$ occurs as a composition factor of the regular module; the Jordan–Hölder theorem for modules makes that independent of the chosen series.

The quotient conclusion holds for every unital algebra. If an arbitrary infinite-dimensional regular module has no finite composition series, the universally valid statement is that $S$ is a simple subquotient; a claim about the factors of a finite series requires finite length.

A simple module is nonzero and has no proper nonzero submodule. A composition series of a module is a finite strict chain $0=V_0<V_1<\cdots <V_n=V$ of submodules with simple quotients $V_i/V_{i-1}$. These quotients are the composition factors of the series.

The Jordan–Hölder theorem for modules says that if a module has a composition series, then any two such series have the same length and the same simple factors up to isomorphism and reordering. In particular, every finite-dimensional module has a well-defined multiset of composition factors.

A composition series of a module is a finite chainof submodules whose factors $M_j/M_{j-1}$ are simple modules. A finite-dimensional module has such a series by induction on its dimension, choosing a maximal proper submodule at each step.

The Jordan–Hölder theorem for modules states that any two composition series of a module have the same length and the same simple factors up to isomorphism and reordering. Thus the multiset of composition factors is an invariant of the module, even when the series itself is not unique.

The flag makes every composition factor one-dimensional, described by a character $\lambda :H\to F$. The Jordan–Hölder theorem for modules for finite-length modules says that the multiset of simple composition factors is independent of the chosen composition series.

Here is a construction valid over every $F$. For $h\in H$, every generalized eigenspace of $\rho (h)$ is an $H$-submodule. Indeed $\mathrm{ad}(h)$ is nilpotent on $H$, say of exponent $c$, andIf $v$ is killed by $(\rho (h)-aI{)}^{N_0}$, take $N\ge N_0+c$. Terms with $j\ge c$ vanish by adjoint nilpotence, and all other terms kill $v$. This proves invariance of the generalized eigenspace.

The flag also makes the characteristic polynomial of every $\rho (h)$ split over $F$. Whenever two composition characters differ, choose an $h$ on which they differ and split the module into the distinct generalized eigenspaces of $\rho (h)$. Repeat within each summand until all its composition factors have the same character. Every nontrivial split lowers dimensions, so the process terminates. Finally combine summands carrying the same character. Jordan–Hölder ensures that the resulting distinct blocks have distinct factor types, givingThis recursive method does not assume that a single element separates every character, which could fail over a finite field.