  The classical construction of the Weil representation, that is with complex
coefficients, has been expected to be working for more general coefficient
rings. This paper exhibits a minimal ring $\mathcal{A}$, being the integral
closure of $\mathbb{Z}[\frac{1}{p}]$ in a cyclotomic field, and carries the
construction of the Weil representation over $\mathcal{A}$-algebras. As a
leitmotiv all along the work, most of the problems can actually be solved over
the based ring $\mathcal{A}$ and transferred for any $\mathcal{A}$-algebra by
scalar extension. The most striking fact consists in these many Weil
representations arising as the scalar extension of a single one with
coefficients in $\mathcal{A}$. In this sense, the Weil module obtained is
universal. Building upon this construction, we are speculating and making
prognoses about an integral theta correspondence.
