Is every (left) finitely generated projective modules over the matrix ring $M_n(\mathbb{C})$ isomorphic to a trivial module? Is there a good reference to look at this problem?
Apologies for asking what is likely a very simple question - note it is really about the isomorphism classes, not K-theory.
The ring $\mathbb{M}_n(\mathbb{C})$ is a semisimple ring, and so every module is a sum of simple modules, and is projective. For this ring, there is only one simple module $S$, up to isomorphism. Thus, every finitely generated module is $S^{(k)}$ for some $k\in \mathbb{N}$. (The module $S$ is isomorphic to an $n\times 1$ column vector, acted on by $R$ by left multiplication.)
You can find a good treatment of these facts in the first 3 chapters of Lam's book "A First Course in Noncommutative Rings".
