relative K1-Group as kernel
Proposition
Let 
be a ring, 
an ideal
Then the relative K1-Group is given by
Proof
follows from the snake lemma for
where we get (after identification of the first two kernels with the 
and 
) the exact sequence
hence it suffices to show that the map
is an epimorphism, since
It is surjective as we may lift any generator 
to 