  The renormalization constant $Z_J$ of the flavor-singlet axial-vector current
with a non-anticommuting $\gamma_5$ in dimensional regularization is determined
to order $\alpha_s^3$ in QCD with massless quarks. The result is obtained by
computing the matrix elements of the operators appearing in the axial-anomaly
equation $\big[\partial_{\mu} J^{\mu}_{5} \big]_{R} = \frac{\alpha_s}{4 \pi}\,
n_f\, {\displaystyle \mathrm{T}_{F}} \, \big[F \tilde{F} \big]_{R}$ between the
vacuum and a state of two (off-shell) gluons to 4-loop order. Furthermore,
through this computation, the conjectured equality between the
$\overline{\mathrm{MS}}$ renormalization constant $Z_{F\tilde{F}}$ associated
with the operator $\big[F \tilde{F} \big]_{R}$ and that of $\alpha_s$ is
verified explicitly to hold true at 4-loop order. This equality automatically
ensures a relation between the respective anomalous dimensions,
$\gamma_{{\scriptscriptstyle J}} = \frac{\alpha_s}{4 \pi}\, n_f\,
{\displaystyle \mathrm{T}_{F}} \, \gamma_{{\scriptscriptstyle FJ}} $, at order
$\alpha_s^4$ given the validity of the axial-anomaly equation which was used to
determine the non-$\overline{\mathrm{MS}}$ piece of $Z_J$ for the particular
$\gamma_5$ prescription in use.
