Let $D$ be a division ring. For $n \ge 1$, let $GL_n(D)$ be the group of invertible $n \times n$ matrices over $D$, let $GL(D)=\varinjlim_n GL_n(D)$ under the stabilization maps $A\mapsto \operatorname{diag}(A,1)$, and let $E(D)\trianglelefteq GL(D)$ be the stable elementary subgroup generated by all elementary transvections $e_{ij}(r)=I_n+rE_{ij}$ with $n\ge2$, $1\le i\ne j\le n$, and $r\in D$, where $I_n$ is the identity matrix and $E_{ij}$ is the matrix unit. Define $K_1(D)=GL(D)/E(D)$. Let $[D^\times,D^\times]$ be the multiplicative commutator subgroup of the unit group $D^\times$, and let $\pi:D^\times\to D^\times/[D^\times,D^\times]$ be the quotient homomorphism. Let $\delta_n:GL_n(D)\to D^\times/[D^\times,D^\times]$ denote the Dieudonne determinant, characterized by $\delta_n(e_{ij}(r))=1$, $\delta_n(\operatorname{diag}(a_1,\dots,a_n))=\pi(a_1\cdots a_n)$, and compatibility with stabilization. Then the induced stable Dieudonne determinant defines a group isomorphism

\begin{align*} \det_D:K_1(D)\xrightarrow{\cong}D^\times/[D^\times,D^\times]. \end{align*}