  We investigate the relationship between ideal membership of an operator and
its pieces relative to several canonical types of partitions of the entries of
its matrix representation with respect to a given orthonormal basis. Our main
theorems establish that if $T$ lies in an ideal $\mathcal{I}$, then $\sum P_n T
P_n$ (or more generally $\sum Q_n T P_n$) lies in the arithmetic mean closure
of $\mathcal{I}$ whenever $\{P_n\}$ (and also $\{Q_n\}$) is a sequence of
mutually orthogonal projections; and in any basis for which $T$ is a block band
matrix, in particular, when in Patnaik--Petrovic--Weiss universal block
tridiagonal form, then all the sub/super/main-block diagonals of $T$ are in
$\mathcal{I}$. And in particular, the principal ideal generated by this $T$ is
the finite sum of the principal ideals generated by each sub/super/main-block
diagonals.
