I am under the impression that in the definition of the Grothendieck group $K_0(R)$ of a (non-commutative) ring it doesn't matter whether we apply the usual $K_0$ construction to the exact category of all finitely generated projective left $R$-modules, or if we apply the construction to the category of all finitely generated projective right $R$-modules.
Is this obvious? To be honest my first reaction is disbelief but given that certain other notions for rings (such as semisimplicity of a ring or Morita equivalence for rings) are left/right-independent, I guess I can believe it. But it is not clear to me why this is true in the case of $K_0$.
I've taken a look in the standard references for classical algebraic $K$-theory but none of them seem to mention this point and either deal just with left $R$-modules throughout, or make no mention at all of the handedness of their modules.
This circle of ideas leads to a more general and vague follow up question: Does anyone have any intuition for why certain notions for non-commutative rings (such as the examples mentioned above: semisimplicity, Morita equivalence, $K_0$) do not depend on whether we look at properties of the collection of left modules over the ring or whether we look at properties of the collection of right modules over the ring.