Let $R$ be a commutative unital ring. Let $GL(R)=\varinjlim_n GL_n(R)$ under the stabilization maps $A\mapsto \operatorname{diag}(A,1)$, let $E(R)\le GL(R)$ be the stable elementary subgroup, and let $K_1(R)=GL(R)/E(R)$, written additively. Let $q:GL(R)\to K_1(R)$ be the quotient map, so the addition in $K_1(R)$ is induced by multiplication in $GL(R)$. Let $\det:K_1(R)\to R^\times$ be the stable determinant homomorphism induced by the usual determinants $\det_n:GL_n(R)\to R^\times$, where $R^\times$ is viewed as an abelian group under multiplication. Let $s:R^\times\to K_1(R)$ be the homomorphism defined by $s(u)=q((u))$, where $(u)\in GL_1(R)\subset GL(R)$ is the $1\times 1$ invertible matrix with entry $u$. Then $\det\circ s=\operatorname{id}_{R^\times}$. Consequently $s$ is injective and $K_1(R)\cong R^\times\oplus\ker(\det)$ as abelian groups. In particular, $R^\times$ is a direct summand of $K_1(R)$.