On purely inseparable extensions $k[X,Y]/k[X',Y']$ and their generators

Let ${\mathbf{k}}$ be a field of characteristic $p>0$ and $R={\mathbf{k}} [X,Y]$ a polynomial algebra in two variables. By a $p$-generator of $R$ we mean an element $u$ of $R$ for which there exist $v\in R$ and $n\ge0$ such that ${\mathbf{k}} [u,v]\supseteq R^{p^n}$. We also define a $p$-line of $R$ to mean any element $u$ of $R$ whose coordinate ring $R/uR$ is that of a $p$-generator. Then we prove that if $u\in R$ is such that $u-T$ is a $p$-line of ${\mathbf{k}} (T)[X,Y]$ (where $T$ is an indeterminate over $R$), then $u$ is a $p$-generator of $R$. This is analogous to the well-known fact that if $u\in R$ is such that $u-T$ is a line of ${\mathbf{k}} (T)[X,Y]$, then $u$ is a variable of $R$. We also prove that if $u$ is a $p$-line of $R$ for which there exist $\varphi\in\operatorname{qt} R$ and $n\ge0$ such that ${\mathbf{k}} (u,\varphi)\supseteq R^{p^n}$, then $u$ is in fact a $p$-generator of $R$.