  We establish a version "over the ring" of the celebrated Hilbert
Irreducibility Theorem. Given finitely many polynomials in $k+n$ variables,
with coefficients in $\mathbb Z$, of positive degree in the last $n$ variables,
we show that if they are irreducible over $\mathbb Z$ and satisfy a necessary
"Schinzel condition", then the first $k$ variables can be specialized in a
Zariski-dense subset of ${\mathbb Z}^k$ in such a way that irreducibility over
${\mathbb Z}$ is preserved for the polynomials in the remaining $n$ variables.
The Schinzel condition, which comes from the Schinzel Hypothesis, is that, when
specializing the first $k$ variables in ${\mathbb Z}^k$, the product of the
polynomials should not always be divisible by some common prime number. Our
result also improves on a "coprime" version of the Schinzel Hypothesis: under
some Schinzel condition, coprime polynomials assume coprime values. We prove
our results over many other rings than $\mathbb Z$, e.g. UFDs and Dedekind
domains for the last one.
