Let $A$ be a two dimensional commutative associative algebra over field $K$ of reals or complex numbers.
Assume that $A$ has units $e$. Let $u \notin Ke$. Then $\{e,u\}$ is basis of $A$. In order to determine that algebra it suffices to know $u\cdot u$. Let $u=pu+qe$. Let's consider polynom $f(x)=x^2-px+q \in K[x]$. Three cases may occur: $f$ has two, one or none roots. In the first case putting $v=(y_2-y_1)^{-1}(u-y_1)$, where $y_1,y_2$ are roots of $f$, we have $v^2=v$ and $\{e,v\}$ is the basis of $A$. In the second putting $v=u-y_1e$, where $y_1$ is a root of $f$, we have $v^2=0$ and ${e,v}$ is the basis of $A$. In the third case $A$ is a field.
How to determine all two dimensional commutative associative algebras without units?
Thanks.