  We investigate the Grassmann envelope (of finite rank) of a
finite-dimensional $\mathbb{Z}_2$-graded algebra. As a result, we describe the
polynomial identities of $G_1(\mathcal{A})$, where $G_1$ stands for the
Grassmann algebra with $1$ generator, and $\mathcal{A}$ is a
$\mathbb{Z}_2$-graded-simple associative algebra. We also classify the
conditions under which two associative $\mathbb{Z}_2$-graded-simple algebras
share the same set of superpolynomial identities, i.e., the polynomial
identities of its Grassmann envelope (in particular, of finite rank). Moreover,
we extend the construction of the Grassmann envelope for the context of
$\Omega$-algebras and prove some of its properties. Lastly, we give a
description of $\mathbb{Z}_2$-graded-simple $\Omega$-algebras.
