A module has a composition series if and only if it is Noetherian and Artinian, the converse using dependent choice

Let $0=M_0<\cdots <M_n=M$ be a composition series [L1] and induct on $i$ that $M_i$ is Noetherian and Artinian. The zero module $M_0$ satisfies both conditions vacuously. A simple factor has only the submodules $0$ and itself; any one nonzero element generates it, since its cyclic submodule is nonzero. Thus all its submodules are finitely generated and all descending chains stabilize. Applying [L4] to $0\to M_{i-1}\to M_i\to M_i\mathrm{/}{M}_{i-1}\to 0$ carries both conditions to $M_i$. At $i=n$ this gives the forward implication, without any choice principle.

Conversely, assume dependent choice and let $M$ be Noetherian and Artinian. Every submodule of $M$ is Noetherian by [L4], so a nonzero submodule $N$ has a maximal proper submodule by the nonempty maximal condition of [L2], using DC. Define a relation on the submodules of $M$ by taking a maximal proper submodule at each nonzero term and letting zero be its own successor. This relation is serial. DC, available from the assumed AC, supplies a sequence starting at $M$; it strictly decreases until it reaches zero and is then constant. These are the precise choice uses in the converse.

The chain of step 1.2 is strictly descending while its terms are nonzero, so the descending chain condition forces some $N_r=0$. Since $N_{k+1}$ is maximal proper in $N_k$, the quotient $N_k\mathrm{/}{N}_{k+1}$ is nonzero and has no proper nonzero submodule, hence is simple. Reversing the chain gives $0=N_r<\cdots <N_0=M$, a composition series. For $M=0$ the empty chain is already the required series, so no choice is consumed in that case. This proves the stated claim.