  We use cosmological hydrodynamical simulations including star formation and
metal enrichment to study the evolution of the chemical properties of
galaxy-like objects at high redshift in the range $0.25<z< 2.35$ in a
hierarchical clustering scenario. As the galactic objects are assembled we find
that their gaseous components exhibit neutral Hydrogen column densities with
abundances and scatter comparable to those observed in damped Lyman-$\alpha$
systems (DLAs).The unweighted mean of abundance ratios and least square linear
regressions through the simulated DLAs yield intrinsic metallicity evolution
for the [Zn/H] and [Fe/H], consistent with results obtained from similar
analysis of available observations. Our model statistically reproduces the mild
evolution detected in the metallicity of the neutral hydrogen content of the
Universe, given by mass-weighted means,if observational constraints are
considered (as suggested by Boiss\'ee et al. 1998). For the $\alpha$-elements
in the simulated DLAs, we find neither enhancement nor dependence on
metallicity. Our results support the hypotheses that DLAs trace a variety of
galactic objects with different formation histories and that both SNI and SNII
are contributing to the chemical enrichment of the gas component at least since
$z \approx 2$. This study indicates that DLAs could be understood as the
building blocks that merged to form today normal galaxies within a hierarchical
clustering scenario.
