  In algebraic number theory, the finiteness of the Picard group of an order in
a number field is generally proved via a lattice argument: the order forms a
lattice and every ideal class contains an integral ideal with a small enough
non-zero element. In this work we will consider $\overline{\mathbb{Z}}$, the
ring of algebraic integers, which is a lattice in a similar sense, and we will
treat this lattice as intrinsically interesting. We will prove several
properties of $\overline{\mathbb{Z}}$ and state some open problems. Many of
these properties have connections to the indecomposable elements of the
lattice, and our main result regards the decidability of the question whether
an algebraic integer is indecomposable.
