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...

Thank you

positiveproperty rather than just non(power)associativity and noncommutativity, it might be of some use. Negative properties hardly interest anyone here (which is the reason why people misunderstood your "nonassociative" as "not necessarily associative" first!). – darij grinberg Nov 30 '11 at 1:56