In III.4 (p.220) it is asserted that
Z\ast,k = Z[\sqrt{|k|},\sqrt{|k|}-1] is a principal ideal domain
(and that therefore submodules of free modules are free). As A.Neeman
and C.Geiss have pointed out, this is nonsense!
For example,
for |k|=5, one gets two different factorizations
4 = 2\cdot 2 = (\sqrt{5}+1)(sqrt{5}-1).