For each unital ring $A$, let $V(A)$ denote the commutative monoid of isomorphism classes of finitely generated projective left $A$-modules under direct sum, let $K_0(A)$ denote the Grothendieck group completion of $V(A)$, and let $\iota_A:V(A)\to K_0(A)$ denote the canonical monoid homomorphism. Write $[P]$ also for $\iota_A([P])$ when no confusion can arise. Let $R$ and $S$ be unital rings, and let $\varphi:R\to S$ be a unital ring homomorphism. Regard $S$ as an $(S,R)$-bimodule by its usual left $S$-module structure and by the right $R$-action

\begin{align*} s\cdot r=s\varphi(r) \end{align*}

for $s\in S$ and $r\in R$. Then extension of scalars sends finitely generated projective left $R$-modules to finitely generated projective left $S$-modules and induces a unique [group homomorphism](/page/Group%20Homomorphism)

\begin{align*} K_0(\varphi):K_0(R)\to K_0(S) \end{align*}

satisfying

\begin{align*} K_0(\varphi)(\iota_R([P]))=\iota_S([S\otimes_R P]) \end{align*}

for every finitely generated projective left $R$-module $P$, where $S\otimes_R P$ carries the left $S$-module structure induced by multiplication on the first tensor factor. For every unital ring $R$,

\begin{align*} K_0(\operatorname{id}_R)=\operatorname{id}_{K_0(R)}. \end{align*}

Moreover, if $R$, $S$, and $T$ are unital rings and $\varphi:R\to S$ and $\psi:S\to T$ are unital ring homomorphisms, then

\begin{align*} K_0(\psi\circ\varphi)=K_0(\psi)\circ K_0(\varphi) \end{align*}

as group homomorphisms from $K_0(R)$ to $K_0(T)$.