  The gauge invariant and anomaly-free 331 model is studied with $R_\xi$ gauges
in order to see which of the right-handed singlet between the case (i),
assigning the bosonic singlet to the counterpart of 3 and $\bar 3$
representations, and the case (ii), assigning fermionic siglet to the
counterpart of the left-handed 3 representation and assigning the antifermionic
singlet to that of the left-handed $\bar 3$ representaion, should be assigned
in physics but not mathematics to the counterparts of the left-handed
fundamental 3 representation and the left-handed complex conjugate $\bar 3$
representation. It is shown that through an effect on the Yukawa interaction of
the transformation property of the singlet it depends on the choices of the
case (i) or the case (ii) whether the anomaly coefficients even after the
spontaneous symmetry breaking (SSB) vanish or not, and the coefficients vanish
only in the case (ii) but do not in the case (i). Furthermore, it is pointed
out that the BRS invariance after the SSB holds for the case (ii) and (i-b) of
two possibilities in the choice for the mass eigenstates in the case (i) and
thus the BRS invariance does not necessarily ensure renormalizability of the
theory though the inverse of it holds.
