= Scalar action obstructs a cyclic module generator
{title2=$Av=Kv\text{ for all }v,\quad\dim_KV>1\implies V\text{ not cyclic}$}
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>.
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