Is the following statement interesting or even trivial ?
- For every $n$ - dimensional associative algebra $A$ over a field $F$ there is a $n +1$ - dimensional nonassociative algebra $V_A$ over $F$ with the following properties :
$1.$ $V_A$ is non commutative and non power associative !
$2.$ $A$ is isomorphic to $N(V_A)$, where $N(V_A)$ is the nucleus of $V_A$.
$3.$ If $n +1$ is odd then $Z(V_A ) = N(V_A)$, where $Z(V_A)$ is the center of $V_A$.
Ps - Sorry guys I have changed the formulation few times.The last change was due to a typo, I meant $V_A$ to be of dimension $n + 1$ ! I will now stop, thanks for all the replies...