Stable tameness of coordinates in two variables (Q1730195)

Consider $z$-automorphisms (i.e. sending $z$ to $z$) of $k[x,y,z]$. It has been proved that they are stably tame in the famouse paper ``Stable tameness of two-dimensional polynomial automorphisms over a regular ring'' by Joost Berson, Arno van den Essen, David Wright [\textit{J. Berson} et al., Adv. Math. 230, No. 4--6, 2176--2197 (2012; Zbl 1246.14075)]. In fact, $k$ can be replaced by regular ring, instead of one variable $z$ one can consider any set $z_1,\dots,z_n$. \par A non-commutative version of this result was obtained in the paper [Sel. Math., New Ser. 18, No. 4, 799--802 (2012; Zbl 1268.16025)] by \textit{A. Belov-Kanel} and \textit{J.-T. Yu}. \par The paper under review is devoted to a coordinate analog of the Joost Berson, Arno van den Essen, David Wright result. It was proven that for a given commutative ring $R$ with unity, every coordinate of $R^2$ is stably tame ($R^2=R[x,y]$ by definition as usual). \par The method of proof is based on localization and factorisation technique used in module theory. The paper is clear and nicely written.