We show that if $R$ is a right Noetherian ring containing the rational numbers, any bound on the Goldie dimension of prime factors of $R[x]$ is actually a bound on the Goldie dimension of prime factors of any polynomial ring $R[x_1,\dots,x_m]$. A result of Sigurdsson then shows that the same bound holds for any iterated differential operator ring $R[\theta;\delta_1]\dots[\theta;\delta_m]$.
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