symmetric monoidal (∞,1)-category of spectra
A filtered ring (resp. filtered algebra) is a monoid object in the category of filtered abelian groups (resp. filtered vector space?s).
One considers positive and negative filtrations, as well as $\mathbb{Z}$-filtrations.
To-do list: complete filtrations, associated graded ring, symbol map, Poisson structure on the associated graded algebra if the latter is commutative.
A major example is the universal enveloping algebra of any Lie algebra.
See also Lazard's criterion and microlocalization.