  P. Aluffi introduced in [1] a new graded algebra in order to conveniently
express characteristic cycles in the theory of singular varieties. This algebra
is attached to a surjective ring homomorphism $A\surjects B$ by taking a
suitable inverse limit of graded algebras, one for each representation of $A$
as a residue ring of a given "ambient" ring $R.$ Since giving a ring surjection
$A\surjects B$ is tantamount to giving an ideal $I \subset A$, it would seem
natural to ask for an analogous notion for $A$-modules. This is the central
purpose of this work. Since a given module may not admit any embedding into a
free module, a preparatory toil includes dealing with this technical point at
the outset. On the bright side, the intrusion of modules raises a few algebraic
questions interesting on their own. It is to expect that this extension to
modules may be transcribed in terms of coherent sheaves, thus possibly
providing an answer to a question by Aluffi in this regard. Two main bodies of
examples are treated in detail to illustrate how the theory works and to show
the relation to finer properties of other algebras.
