Scalar action obstructs a cyclic module generator ($Av=Kv\text{ for all }v,{\dim }_{K}V>1⟹V\text{ not cyclic}$)

If an algebra acts on a vector space only by scalar multiples of the identity, every vector generates at most a one-dimensional submodule. A higher-dimensional such module is therefore not cyclic. For example upper triangular $n$-by-$n$ matrices act by the first diagonal entry on their first-row ideal. For $n>1$ this ideal is a noncyclic submodule of the cyclic regular module. Thus a submodule of a cyclic module need not be cyclic, even over a finite-dimensional algebra.