We present results from SPH-cosmological simulations, including self-consistent modeling of supernova feedback and chemical evolution, of galaxies belonging to two clusters and 12 groups. We reproduce the mass-metallicity (ZM) relation of galaxies classified in two samples according to their star-forming (SF) activity, as parameterized by their specific star formation rate (sSFR), across a redshift range up to z = 2. The overall ZM relation for the composite population evolves according to a redshift-dependent quadratic functional form that is consistent with other empirical estimates, provided that the highest mass bin of the brightest central galaxies is excluded. Its slope shows irrelevant evolution in the passive sample, being steeper in groups than in clusters. However, the subsample of high-mass passive galaxies only is characterized by a steep increase of the slope with redshift, from which it can be inferred that the bulk of the slope evolution of the ZM relation is driven by the more massive passive objects. The scatter of the passive sample is dominated by low-mass galaxies at all redshifts and keeps constant over cosmic times. The mean metallicity is highest in cluster cores and lowest in normal groups, following the same environmental sequence as that previously found in the red sequence building. The ZM relation for the SF sample reveals an increasing scatter with redshift, indicating that it is still being built at early epochs. The SF galaxies make up a tight sequence in the SFR-M * plane at high redshift, whose scatter increases with time alongside the consolidation of the passive sequence. We also confirm the anti-correlation between sSFR and stellar mass, pointing at a key role of the former in determining the galaxy downsizing, as the most significant means of diagnostics of the star formation efficiency. Likewise, an anti-correlation between sSFR and metallicity can be established for the SF galaxies, while on the contrary more active galaxies in terms of simple SFR are also metal-richer. Finally, the [O/Fe] abundance ratio is presented too: we report a strong increasing evolution with redshift at given mass, especially at z ≳ 1. The expected increasing trend with mass is recovered when only considering the more massive galaxies. We discuss these results in terms of the mechanisms driving the evolution within the high- and low-mass regimes at different epochs: mergers, feedback-driven outflows, and the intrinsic variation of the star formation efficiency.