  We prove that every Ariki-Koike algebra is Morita equivalent to a direct sum
of tensor products of smaller Ariki-Koike algebras which have q-connected
parameter sets. A similar result is proved for the cyclotomic q-Schur algebras.
Combining our results with work of Ariki and Uglov, the decomposition numbers
for the Ariki-Koike algebras defined over fields of characteristic zero are now
known in principle.
