  We investigate modules over "systematic" rings. Such rings are "almost
graded" and have appeared under various names in the literature; they are
special cases of the G-systems of Grzeszczuk. We analyse their K-theory in the
presence of conditions on the support, and explain how this generalises and
unifies calculations of graded and filtered K-theory scattered in the
literature. Our treatment makes systematic use of the formalism of idempotent
completion and a theory of triangular objects in additive categories, leading
to elementary and transparent proofs throughout.
