Let $R$ and $S$ be unital rings, and put $A:=R\times S$. For every unital ring $T$, let $V(T)$ be the commutative monoid of isomorphism classes of finitely generated projective left $T$-modules under direct sum, and let $K_0(T)$ be its Grothendieck group completion. For each integer $n\geq 1$, let $M_n(T)$ be the ring of $n\times n$ matrices over $T$, let $I_n\in M_n(T)$ be the identity matrix, and let $GL_n(T)=M_n(T)^\times$. Let $GL(T)=\varinjlim_n GL_n(T)$ under the stabilization maps $X\mapsto \operatorname{diag}(X,1_T)$. For $n\geq 2$, $1\leq i\ne j\leq n$, let $E_{ij,n}\in M_n(T)$ denote the matrix unit with $1_T$ in the $(i,j)$-entry and $0_T$ elsewhere, and for $t\in T$ let $e_{ij,n}(t):=I_n+tE_{ij,n}\in GL_n(T)$. Let $E(T)\trianglelefteq GL(T)$ be the stable elementary subgroup generated by the images of all $e_{ij,n}(t)$. Define $K_1(T):=GL(T)/E(T)$. Then the product projections $A\to R$ and $A\to S$ induce canonical isomorphisms of abelian groups

\begin{align*} K_0(R\times S)\cong K_0(R)\oplus K_0(S) \end{align*}

and

\begin{align*} K_1(R\times S)\cong K_1(R)\oplus K_1(S). \end{align*}