Chang, Gyu Whan and Smertnig, Daniel:
Factorization in the Self-Idealization of a PID
Bollettino dell'Unione Matematica Italiana Serie 9 6 (2013), fasc. n.2, p. 363-377, (English)
pdf (313 Kb), djvu (148 Kb). | MR 3112984 | Zbl 1283.13015
Sunto
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 $R(D)$.
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