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Let $A$ be a finite dimensional algebra over the field $k$ i.e.:

  • $A$ is a commutative associative unital ring.
  • there is a homomorphism $\varphi:k\to A$, which defines a map $k\times A\to A$ as $\alpha\cdot a:=\varphi(\alpha)a$.
  • $\exists\{e_1,\ldots ,e_n\}\subset A$ such that $A=\sum_{i=1}^nk\cdot e_i$.

Am I right that:

  • $A$ is a vector space, in particular $A$ has a basis and $\dim_kA\leq n$,
  • any submodule of $A$ is a finite dimensional vector space?

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