Factorization in the Self-Idealization of a PID

Chang, Gyu Whan ; Smertnig, Daniel

Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 2, pp. 363-377

Résumé

Let $D$ be a principal ideal domain and $R(D) = \{(\begin{smallmatrix} a b \\ 0 a \end{smallmatrix}) \mid a, b \in D\}$ 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 $R(D)$.