Direct sum decompositions of modules, almost trace ideals, and pullbacks of monoids

We show that a number of pullback diagrams appear naturally in the study of pre-ordered Grothendieck groups. The passage of projective modules from a ring R to a factor ring R/I turns out to be particularly good for a certain class of ideals, which we call almost trace ideals. We generalize to arbitrary rings a result by Goodearl concerning the lattice of the directed convex subgroups of K 0 (R). Finally, we show that a variant K 0 ′(I) of the Grothendieck group of I, introduced by Quillen, has an easy description in terms of projective modules when I is an almost trace ideal. © de Gruyter 2006.