Factorization in the self-idealization of a PID

Let $D$ be a principal ideal domain and $R(D) = {\begin{pmatrix} a & b 0 & a \end{pmatrix} \mid a, b \in D}$beitsself-idealization\mathrm{.}Itisknownthat$R(D)$isacommutativenoetherianringwithidentity,andhence$R(D)$ 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 $R(D)$.