Let
k
\mathbf {k}
be a field of characteristic
p
>
0
p>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
.