The Grothendieck group of
finite-length inner product modules over a PID is here shown to be a sum of
countably many copies of the corresponding groups for the residue fields. It follows
that nonsingular pairs of symmetric bilinear forms in characteristic 2 owe their extra
complexity only to lack of a cancellation theorem: The invariants for isometry in
other characteristics continue to determine classes in the Grothendieck group. This is
also true for singular pairs.