Let \(R\) be a ring, \(\mathfrak{a} \subseteq R\) an ideal
Then the relative infinite elementary linear group is defined as kernel
\begin{align*}
E(R, \mathfrak{a}) := \mathrm{ker}(\pi_*: E(R \times_{R/\mathfrak{a}} R) \rightarrow E(R))
\end{align*}
for the infinite elementary linear group functor