TITLE: Is K theory ever trivial because of the ring, and not because of the kinds of modules we look at?
QUESTION [11 upvotes]: Let $\mathcal C$ be some Waldhausen category; we know that the K-theory $K(\mathcal C)$ might be trivial if $\mathcal C$ contains objects that are too ``big'' in some sense---for instance, via the Eilenberg swindle. 
(Example: An infinite-dimensional vector space is isomorphic to a direct sum of itself and a finite-dimensional vector space, so any finite-dimensional and any infinite-dimensional vector space have the same K-theory class; on the other hand, any infinite-dimensional vector space represents an idempotent element in the $K_0$ group, but the only idempotent in a group is the identity, so must represent zero.)
So I have two related questions:
(1) In known examples where the K-theory of $\mathcal C$ is trivial, does it tend to be because of Eilenberg-swindle-type reasons? 
(1)(a) Specializing to the categories of sheaves on a variety, or on a non-commutative space, I understand that adjectives like quasi-coherent sheaves, or perfect sheaves, or coherent sheaves, or Ind-coherent sheaves, are important to distinguish how big the objects in our categories are. If you would be able to clarify the size or `infinite-dimensionality' of the objects in each such category, I would be very grateful! For instance, should I just always assume that the stable oo-category of Ind-coherent sheaves always has trivial K-theory?
(2) Are there examples of when the underlying `ring' is so big that the K-theory has to be trivial? For instance, if I have some non-compact space with gigantic $\pi_1$, is it true that if I looked at its based loop space and modules over the based loop space, its K-theory will be trivial? What if I only look at perfect modules over such an algebra? (Here, you can take singular chains to think about based loops as an algebra in chain complexes, or you can think of it as an $E_1$ ring spectrum. I am interested in either answer.)

REPLY [10 votes]: If $R\cong R\oplus R$ as an $R$-bimodule, then for any module $R$-module $N$, we have
$$N\cong N\otimes R\cong N\otimes (R\oplus R)\cong (N\otimes R)\oplus 
(N\otimes R)\cong N\oplus N$$
so that every $[N]$ is trivial in $K$-theory.
For a concrete example, let $S$ be any ring and $R$ the ring of column-finite countable-by-countable matrices over $S$.