  Let $D$ be a principal ideal domain and $R(D) = \{\begin{pmatrix} a & b 0 & a
\end{pmatrix} \mid a, b \in D\}$ be its self-idealization. It is known that
$R(D)$ is a commutative noetherian ring with identity, and hence $R(D)$ is
atomic (i.e., every nonzero nonunit can be written as a finite product of
irreducible elements). In this paper, we completely characterize the
irreducible elements of $R(D)$. We then use this result to show how to
factorize each nonzero nonunit of $R(D)$ into irreducible elements. We show
that every irreducible element of $R(D)$ is a primary element, and we determine
the system of sets of lengths of $R(D)$.
