forgetful natural transformation from relative K1 to K1 of rings

Let \(R\) be a ring, \(\mathfrak{a} \subseteq R\) an ideal
Then there exists a morphism

\begin{align*} K_1(R, \mathfrak{a}) \rightarrow K_1(R) \end{align*}

natural with respect to category of ideal pairs of rings induced by

\begin{align*} \mathrm{GL}(R, \mathfrak{a}) \rightarrow& \mathrm{GL}(R) \\ (1,B) \mapsto& B \end{align*}

see:

• relative infinite general linear group
  

omitted