  This paper extends the well-known Ringrose theory for compact operators to
polynomially Riesz operators on Banach spaces. The particular case of an
ideal-triangularizable Riesz operator on an order continuous Banach lattice
yields that the spectrum of such operator lies on its diagonal, which motivates
the systematic study of an abstract diagonal of a regular operator on an order
complete vector lattice $E$. We prove that the class $\mathscr D$ of regular
operators for which the diagonal coincides with the atomic diagonal is always a
band in $\mathcal L_r(E)$, which contains the band of abstract integral
operators. If $E$ is also a Banach lattice, then $\mathscr D$ contains positive
Riesz operators.
