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 $K$ ?

If there is an integer $d$ such that, for all $x \in A$, the degree of $P_x$ is less than $d$, it is true.

Thanks in advance.


This is the Kurosh problem, which has a negative solution. If I recall correctly, one exhibits an example using the Golod-Shafarevich lemma.

Wikipedia has a page on this, in fact. The example was constructed by Golod.


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