Ideal structure of $C^{\ast }$-algebras associated with $C^{\ast }$-correspondences

We study the ideal structure of $C^{\ast }$-algebras arising from $C^{\ast }$-correspondences. We prove that gauge-invariant ideals of our $C^{\ast }$-algebras are parameterized by certain pairs of ideals of original $C^{\ast }$-algebras. We show that our $C^{\ast }$-algebras have a nice property which should be possessed by generalization of crossed products. Applications to crossed products by Hilbert $C^{\ast }$-bimodules and relative Cuntz-Pimsner algebras are also discussed.