relative infinite general linear group depends only on the ideal

Let \(R,S\) be rings, \(\mathfrak{a} \subseteq R, \mathfrak{b} \subseteq S\) be ideals and \(f: R \rightarrow S\) a ring-homomorphism such that

\begin{align*} f_{\vert \mathfrak{a}}: \mathfrak{a} \rightarrow \mathfrak{b} \end{align*}

is a Bijection

Then

\begin{align*} \mathrm{Gl}(R, \mathfrak{a}) \cong \mathrm{Gl}(S, \mathfrak{b}) \end{align*}

follows from

• relative infinite general linear group and projection