On the uniform dimension of skew polynomial rings in many variables

Let $R$ be an associative ring, $X=\{x_i\colon\ i\in\Gamma\}$ be a nonempty set of variables, $F=\{f_i\colon\ i\in\Gamma\}$ be a family of injective ring endomorphisms of $R$ and $A(R,F)$ be the Cohn–Jordan extension. In this paper we prove that the left uniform dimension of the skew polynomial ring $R[X,F]$ is equal to the left uniform dimension of $A(R,F)$, provided that $Aa\ne0$ for all nonzero $a\in A$. Furthermore, we show that for semiprime rings the equality $\dim R=\dim R[X,F]$ does not hold in the general case. The following problem is still open. Does $\dim R=\dim R[x,f]$ hold if $R$ is a semiprime ring, $f$ is an injective ring endomorphism of $R$ and $\dim R<\infty$?