Let $K$ be a field, and $A$ a finitely generated associative algebra over $K$. We suppose that $A$ has a unit and that every element $x$ of $A$ is annihilated by a non-zero polynomial $P\_x$ depending on $x$. Is $A$ a finite-dimensional vector space over $K$ ?


Under the additional assumption that there is an integer $d$ such that, for all $x \in A$, the degree of $P\_x$ is less than $d$, is the result true?



Finitely generated algebra in which every element is annihilated by a non-zero polynomial
This is the Kurosh problem, which has a negative solution. If I recall correctly, one exhibits an example using the Golod-Shafarevich lemma.


[Wikipedia](http://en.wikipedia.org/wiki/Kurosh_problem) has a page on this, in fact. The example was constructed by Golod.


