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?