  We discuss $Z$ -production in the deep inelastic process $e + P \to e + Z +
X$ in the context of the Standard Model using the Parton Model. In contrast to
the deep inelastic $eP$-scattering ($e + P \to e + X$), where the choice of
$\tilde{Q}^{2} $ is unambiguous, in this case is not clear since the momentum
transfer square, at the quark level, depends on the reaction mechanism. Our aim
in this work is to show that the total cross section depends strongly on the
prescription used for the scale parameter $\tilde{Q}^{2}$ and on a different
option that we have taken for making the convolution of the parton distribution
functions and the amplitude of the quark processes. We present results for the
total cross section as a function of the total energy $\sqrt{s}$ of the system
$eP$, in the range $300 \leq \sqrt{s} \leq 1300$ GeV. We find differences of up
to 25\% in the rates of the total cross section for the different prescriptions
that we have taken, for $\sqrt{s} \approx 1300$ GeV, which is the expected
maximal total energy of the system $eP$ to be reached at the LHeC. Taking an
integrated luminosity of $ab^{-1}$, such a difference (25\%) corresponds to
$\approx 10^4$ produced $Z$ bosons. Finally, we propose to modify slightly the
Parton Model in order to do a simple and practical prescription for the
calculation without ambiguities of the $Z$-production in the deep inelastic
$eP$ scattering.
