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An \textit{ideal} of $N$-tuples of operators is a class invariant withrespect to unitary equivalence which contains direct sums of arbitrarycollections of its members as well as their (reduced) parts. New decompositiontheorems (with respect to ideals) for $N$-tuples of closed densely definedlinear 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 establishedand certain ideals in $CDD_N$ are distinguished. It is proved that infiniteoperations in $CDD_N$ may be reconstructed from the direct sum operation of apair. \textit{Prime decomposition} in $CDD_N$ is proposed and its (in a sense)uniqueness is established. The issue of classification of ideals in $CDD_N$ (upto isomorphism) is discussed. A model for $CDD_N$ is described and its concreterealization is presented. A new partial order of $N$-tuples of operators isintroduced and its fundamental properties are established. Extremal importanceof unitary disjointness of $N$-tuples and the way how it `tidies up' thestructure of $CDD_N$ are emphasized.

Unitary equivalence and decompositions of finite systems of closed densely defined operators in Hilbert spaces