K-theory - dependence of algebraic structure: What is the K-theory of a direct product?

Question

I want to figure out the dependence of K-class of an finitely generate projective ring and its algebraic struture.

For example consider $K_0(\mathbb{C})\cong \mathbb{Z}$ and $K_0(\mathbb{R}\times\mathbb{R})$, where the algebraic on $\mathbb{C}$ is the standard multiplication and on $\mathbb{R}\times\mathbb{R}$ the multipation is given by: $(r_1,r_2)\times (r_1',r_2')\mapsto (r_1*r_1',r_2*r_2')$. Are $K_0(\mathbb{C})$ and $K_0(\mathbb{R}\times\mathbb{R})$ isomorphic? What can we say in general?

Answer 1

Presumably algebraic K-theory preserves finite direct sums. In that case $K_0(\mathbb R\times\mathbb R)$ cannot be $\mathbb Z$.

Answer 2

The grothendieck group of a field $k$ is always equal to $\mathbb{Z}$ - the ring of integers. The isomorphism is as follows:

$i: K_0(k) \cong \mathbb{Z}$ with $i([V]):=dim(V)$.

In general there is for arbitrary comutative unital rings $A_j, j=1,..,l$ an isomorphism

D1. $K_i(A_1\oplus \cdots \oplus A_l)\cong K_i(A_1)\oplus \cdots \oplus K_i(A_l)$

for any integer $i\geq 0$, hence $K_0(\mathbb{R}\times \mathbb{R})\cong \mathbb{Z}\times \mathbb{Z}$.

Question: "Are K0(C) and K0(R×R) isomorphic? What can we say in general?"

Answer: The following holds:

$K_0(\mathbb{C})\cong \mathbb{Z} \neq K_0(\mathbb{R}\times \mathbb{R})$.