  We define a generalization $\mathfrak{G}$ of the Grassmann algebra $G$ which
is well-behaved over arbitrary commutative rings $C$, even when $2$ is not
invertible. In particular, this enables us to define a notion of superalgebras
that does not become degenerate in such a setting. Using this construction we
are able to provide a basis of the non-graded multilinear identities of the
free superalgebra with supertrace, valid over any ring.
  We also show that all identities of $\mathfrak{G}$ follow from the Grassmann
identity, and explicitly give its co-modules, which turn out to be
generalizations of the sign representation. In particular, we show that the
co-module is a free $C$-module of rank $2^{n-1}$.
