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The two-dimensional quantum torus is the algebra with invertible generators $X,Y$ and relation $YX=qXY$, for $q\in k^{\times }$. Its basis consists of $X^i{Y}^j$ with $i,j\in \mathbb{Z}$, with the same multiplication rule as the quantum plane. It is a right Noetherian ring: successive adjunction of $X^{-1}$, $Y$, and $Y^{-1}$ satisfies the hypotheses of the noncommutative Hilbert basis theorem.

If an associative unital algebra has a complete norm and separately continuous multiplication, the Uniform boundedness principle applied to $L_x(y)=xy$, for $\Vert x\Vert \le 1$, yields $\Vert xy\Vert \le C\Vert x\Vert \Vert y\Vert$. The operator norm of $L_x$ then defines an equivalent algebra norm: $\Vert x\Vert /\Vert e\Vert \le \Vert L_x\Vert \le C\Vert x\Vert$, and $L_{xy}=L_x{L}_y$ proves submultiplicativity. It normalizes the identity to norm one.

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.

For a finite-dimensional commutative complex algebra, a simple module is a cyclic quotient of the algebra and hence finite-dimensional. Every action map is a module endomorphism, since the algebra is commutative. The complex Schur's lemma makes each action scalar, so every vector subspace is a submodule; simplicity forces dimension one. The finite-dimensional hypothesis cannot simply be deleted: $\mathbb{C}(t)$ acting on itself is a simple module of infinite complex dimension.

For a self-injective finite-dimensional algebra, remove the projective vertices from its Auslander–Reiten quiver. The resulting stable quiver has vertices the nonprojective indecomposable modules. Its translation sends the end term of an almost-split exact sequence to its start term. In particular a quiver of shape $\mathbb{Z}{A}_s/⟨{\tau }^e⟩$ has $es$ vertices.