Finite generation in $C^{\ast }$-algebras and Hilbert $C^{\ast }$-modules

We characterize $C^{\ast }$-algebras and $C^{\ast }$-modules such that every maximal right ideal (resp. right submodule) is algebraically finitely generated. In particular, $C^{\ast }$-algebras satisfy the Dales–.Zelazko conjecture.