# Finite dimensional algebra over field

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?
-

By definition, $A$ is endowed with an action of $k$ that makes it into a vector space, and the phrase "finite dimensional over $k$" means finite dimensional as a $k$-vector space. (Edit: your dimension comment also follows because your third bullet point says $A$ is spanned by $n$ vectors and thus has dimension $\leq n$).
A submodule of $A$ where $A$ is considered as a $k$-module is by definition a $k$-subspace, and hence is finite-dimensional.