Let $R$ be a unital ring. Let $GL(R)$ denote the stable general linear group of $R$, formed from the groups $GL_n(R)$ under block-sum stabilization, and let $E(R) \le GL(R)$ denote the stable elementary subgroup. Then the [quotient group](/theorems/790)
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\begin{align*}
GL(R)/E(R)
\end{align*}
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is abelian. Moreover, for every unital ring homomorphism $\varphi:R\to S$, entrywise application of $\varphi$ induces a [group homomorphism](/page/Group%20Homomorphism)
and these homomorphisms satisfy $(\operatorname{id}_R)_*=\operatorname{id}_{GL(R)/E(R)}$ and $(\psi\circ\varphi)_*=\psi_*\circ\varphi_*$ for all composable unital ring homomorphisms $R\xrightarrow{\varphi}S\xrightarrow{\psi}T$.