  We introduce the notion of quantum $N$-toroidal algebras as natural
generalization of the quantum toroidal algebras as well as extended quantized
GIM algebras of $N$-fold affinization. We show that the quantum $N$-toroidal
algebras are quotients of the extended quantized GIM algebras of $N$-fold
affinization, which generalizes a well-known result of Berman and Moody for Lie
algebras.
