  The parity-preserving $U_A(1)\times U_a(1)$ massive QED$_3$ is ultraviolet
finiteness -- exhibits vanishing $\beta$-functions, associated to the gauge
coupling constants (electric and pseudochiral charges) and the Chern-Simons
mass parameter, and all the anomalous dimensions of the fields -- as well as is
parity and gauge anomaly free at all orders in perturbation theory. The proof
is independent of any regularization scheme and it is based on the quantum
action principle in combination with general theorems of perturbative quantum
field theory by adopting the Becchi-Rouet-Stora (BRS) algebraic renormalization
method in the framework of Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ)
subtraction scheme.
