boundary map from K1 to K0 of rings
Definition
Let 
be a ring, 
an ideal
Then the boundary map of 
and 
is defined as
where
is canonically identified (cf. whitehead theorem for k1 of a ring)
denotes the isomorphism class of the free module with rank 
denotes the clutching construction for the pullback
i.e.
Welldefined
independent of for 
follows as
independent of representatitive: 
we show that the image of 
lies in the kernel.
Then it suffices to show that 
.
kernel as supset: 
Suppose there exist a preimage 
.
Then
defines an isomorphism.
Thus 
or
reduction: 
follows from choosing lifts 
of generators 