  Motivated by Kraji\v{c}ek and Scanlon's definition of the Grothendieck ring
$K_0(M)$ of a first-order structure $M$, we introduce the definition of
$K$-groups $K_n(M)$ for $n\geq0$ via Quillen's $S^{-1}S$ construction. We
provide a recipe for the computation of $K_1(M_R)$, where $M_R$ is a free
module over a PID $R$, subject to the knowledge of the abelianizations of the
general linear groups $GL_n(R)$. As a consequence, we provide explicit
computations of $K_1(M_R)$ when $R$ belongs to a large class of Euclidean
domains that includes fields with at least $3$ elements and polynomial rings
over fields with characteristic $0$. We also show that the algebraic $K_1$ of a
PID $R$ embeds into $K_1(R_R)$.
