  We propose a candidate, which we call the fractional Galois ideal after
Snaith's fractional ideal, for replacing the classical Stickelberger ideal
associated to an abelian extension of number fields. The Stickelberger ideal
can be seen as gathering information about those $L$-functions of the extension
which are non-zero at the special point $s = 0$, and was conjectured by Brumer
to give annihilators of class-groups viewed as Galois modules. An earlier
version of the fractional Galois ideal extended the Stickelberger ideal to
include $L$-functions with a simple zero at $s = 0$, and was shown by the
present author to provide class-group annihilators not existing in the
Stickelberger ideal. The version presented in this article deals with
$L$-functions of arbitrary order of vanishing at $s = 0$, and we give evidence
using results of Popescu and Rubin that it is closely related to the Fitting
ideal of the class-group, a canonical ideal of annihilators. Finally, we prove
an equality involving Stark elements and class-groups originally due to
B\"uy\"ukboduk, but under a slightly different assumption, the advantage being
that we need none of the Kolyvagin system machinery used in the original proof.
