Let $F$ be a field, $Char.(F)=0$ and $n$ be a natural number. We say that an algebra $A$ over $F$ is $n$-power-associative if $n_{th}$ power of any element is well defined. Is it true that there is a natural number $N(m)$ such that if $n> N(m)$ and $A$ is any $m$-dimensional $n$-power-associative algebra then it is $n+1$-power-associative as well? If "Yes" what is known about $N(m)$?
I don't sure if I understood your question. However my answer is the following. Adrian Albert in 1948 showed that over a field of characteristic diferent of 2, 3 and 5, if $A$ is a commutative algebra and x^2x^2=x^3 x (I guest it is that you mean $4$-power-associative) then subalgebra spanned by x is associative, so, x^n is well-defined for all $n$ being a natural number. Dimension is not important.
This seems a strange question. Do you have a specific motivation for asking it? Do you have natural examples of algebras which are $n$-power associative but not $n-1$-power associative (other, of course, than when every $n$-th power is zero)?
- Badges
- Science topic
- Similar topics
- Mathematical Sciences
- Algebra