An ideal of N-tuples of operators is a class invariant with respect to unitary equivalence which contains direct sums of arbitrary collections of its members as well as their (reduced) parts. New decomposition theorems (with respect to ideals) for N-tuples of closed densely defined linear operators acting in a common (arbitrary) Hilbert space are presented. Algebraic and order (with respect to containment) properties of the class $CDD_{N}$ of all unitary equivalence classes of such N-tuples are established and certain ideals in $CDD_{N}$ are distinguished. It is proved that infinite operations in $CDD_{N}$ may be reconstructed from the direct sum operation of a pair. Prime decomposition in $CDD_{N}$ is proposed and its uniqueness (in a certain sense) is established. The issue of classification of ideals in $CDD_{N}$ (up to isomorphism) is discussed. A model for $CDD_{N}$ is described and its concrete realization is presented. A new partial order of N-tuples of operators is introduced and its fundamental properties are established. The importance of unitary disjointness of N-tuples and the way how it 'tidies up' the structure of $CDD_{N}$ are emphasized.