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

On purely inseparable extensions 𝐾[𝑋,𝑌]/𝐾[𝑋’,𝑌’] and their generators