  We introduce fusion algebras with not necessarily positive structure
constants and without identity element. We prove that they are semisimple when
tensored with $\mathbb{C}$ and that their characters satisfy orthogonality
relations. Then we define the proper notion of subrings and factor rings for
such algebras. For certain algebras $R$ we prove the existence of a ring $R'$
with nonnegative structure constants such that $R$ is a factor ring of $R'$. We
give some examples of interesting factor rings of the representation ring of
the quantum double of a finite group. Then, we investigate the algebras
associated to Hadamard matrices. For an $n\times n$-matrix the corresponding
algebra is a factor ring of a subalgebra of
$\mathbb{Z}[{(\mathbb{Z}/2\mathbb{Z})}^{n-2}]$.
