  If O is a reduced operad in symmetric spectra, an O-algebra I can be viewed
as analogous to the augmentation ideal of an augmented algebra. Implicit in the
literature on Topological Andre-Quillen homology is that such an I admits a
canonical (and homotopically meaningful) decreasing O-algebra filtration I >
I^2 > I^3 > ... satisfying various nice properties analogous to powers of an
ideal in a ring.
  In this paper, we are explicit about these constructions. With R a
commutative S-algebra, we study derived versions of the circle product M o_O I,
where M is an O-bimodule, and I is an O-algebra in R-modules. Letting M run
through a decreasing O-bimodule filtration of O itself then yields the
augmentation ideal filtration as above. The composition structure of the operad
induces algebra maps from (I^i)^j to I^{ij}, fitting nicely with previously
studied structure.
  As a formal consequence, an O-algebra map from I to J^d induces compatible
maps from I^n to J^{dn}, for all n. This is an essential tool in the first
author's study of Hurewicz maps for infinite loop spaces, and its utility is
illustrated here with a lifting theorem.
