  In this paper and a forthcoming joint one with Y. Hachimori we study Iwasawa
modules over an infinite Galois extension K of a number field k whose Galois
group G=G(K/k) is isomorphic to the semidirect product of two copies of the
p-adic numbers. After first analyzing some general algebraic properties of the
corresponding Iwasawa algebra, we apply these results to the Galois group of
the p-Hilbert class field over K.
  As a main tool we prove a Weierstrass preparation theorem for certain skew
power series rings. One striking result in our work is the discovery of the
abundance of faithful torsion modules, i.e. non-trivial torsion modules whose
global annihilator ideal is zero. Finally we show that the completed group
algebra with coefficients in the finite field of p elements is a unique
factorization domain in the sense of Chatters.
