# Finitely generated algebra in which every element is annihilated by a non-zero polynomial

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?