Example of finite-dimensional module over a K-algebra without a composition series

Question

Let $A$ be a $K$-algebra. It is well-known that if $A$ is finite-dimensional as a $K$-vector space then every finite-dimensional $A$-module $M$ has a composition series (more generally, the same is true if $A$ is an artinian ring). Suppose now that this is not the case, i.e. that $A$ is not a finite-dimensional algebra. Then there are some $A$-modules without a composition series; but can we find a finite-dimensional $A$-module $M$ with the same property? More specifically,

1. Can you give an example for some $A$?
2. Is it possible to state that for any such $A$ there exists such an $M$?

Answer 1

Any $A$ module over an $K$ algebra is automatically a $K$ vector space. A chain of $A$ submodules automatically becomes a chain of $K$ subspaces.

In light of this, an $n$-dimensional $A$ module clearly has finite composition length at most $n$, even if $A$ was infinite dimensional.