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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 |
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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 $\mathbb{Z}a$). |
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