  We present the complete set of Renormalisation Group Equations (RGEs) at one
loop for the non-exotic minimal U(1) extension of the Standard Model (SM). It
includes all models that are anomaly-free with the SM fermion content augmented
by one Right-Handed (RH) neutrino per generation. We then pursue the numerical
study of the pure B-L model, deriving the triviality and vacuum stability
bounds on an enlarged scalar sector comprising one additional Higgs singlet
field with respect to the SM.
