relative K1-Group of a ring and ideal

Let \(A\) be a ring, \(\mathfrak{a}\) be an ideal
Then the relative \(K_1\)-group \(K_1(R, \mathfrak{a})\) is defined as quotient group

\begin{align*} K_1(R, \mathfrak{a}) := \mathrm{GL}(R, \mathfrak{a}) / E(R, \mathfrak{a}) \end{align*}

where

1. relative infinite general linear group
   
2. infinite elementary linear group