Let $R$ be a unital ring and let $I \trianglelefteq R$ be a two-sided ideal. Let $GL(R)=\varinjlim_n GL_n(R)$ be the stable general linear group under the inclusions $A \mapsto \operatorname{diag}(A,1)$, and let $E(R) \le GL(R)$ be the stable elementary subgroup. Let $\alpha_R:GL(R)\to GL(R/I)$ be induced entrywise by the quotient map $R\to R/I$, and define $GL(R,I):=\ker(\alpha_R)$. Let $\epsilon_R:E(R)\to E(R/I)$ be the restriction of $\alpha_R$, and define $E(R,I):=\ker(\epsilon_R)=E(R)\cap GL(R,I)$. Using the standard plus-construction model $K(R)=BGL(R)^+$ with respect to $E(R)$, the inclusion $GL(R,I)\to GL(R)$ induces a natural group isomorphism $GL(R,I)/E(R,I)\cong \ker(K_1(R)\to K_1(R/I))$. Equivalently, in the low-degree exact sequence for the homotopy fibre $K(R,I)$, this quotient identifies with the image of $K_1(R,I)\to K_1(R)$, not in general with all of $K_1(R,I)$.