1
$\begingroup$

I am looking for an example of a noncommutative and non power associative n - dimensional algebra $A$ with $N(A)=Z(A)$, where $N(A)$ is the nucleus and $Z(A)$ the center. All the examples coming to my mind are algebras with $Z(A)\subseteq N(A)$

Thank you

$\endgroup$
2
$\begingroup$

I think the following example works. Take an algebra $A$ (say, over $\mathbb{Z}$) with basis $\{ a, b, c \}$ and with products defined by putting $cb = c^2 = b$, and all other products of basis elements equal to $a$. Then $(cc)c = bc = a$, while $c(cc) = cb = c$, so $A$ is not power-associative and non-commutative. But the centre and nucleus are equal (to $\mathbb{Z}a$).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.