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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 $R(D)$ isatomic (i.e., every nonzero nonunit can be written as a finite product ofirreducible elements). In this paper, we completely characterize theirreducible elements of $R(D)$. We then use this result to show how tofactorize each nonzero nonunit of $R(D)$ into irreducible elements. We showthat every irreducible element of $R(D)$ is a primary element, and we determinethe system of sets of lengths of $R(D)$.

Factorization in the self-idealization of a PID