Geodesic
Factorization in the Self-Idealization of a PID
Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 2, pp. 363-377.

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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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Chang, Gyu Whan; Smertnig, Daniel. Factorization in the Self-Idealization of a PID. Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 2, pp. 363-377. http://geodesic.mathdoc.fr/item/BUMI_2013_9_6_2_a5/

[1] D. Adams - R. Ardila - D. Hannasch - A. Kosh - H. Mccarthy - V. Ponomarenko - R. Rosenbaum, Bifurcus semigroups and rings, Involve, 2 (2009), 351-356. | DOI | MR | Zbl

[2] D. D. Anderson - S. Valdes-Leon, Factorization in commutative rings with zero divisors, Rocky Mountain J. Math. 26 (1996), 439-480. | DOI | MR | Zbl

[3] D. D. ANDERSON (ed.), Factorization in Integral Domains, Lect. Notes Pure Appl. Math., vol. 189, Marcel Dekker, 1997. | MR

[4] D. D. Anderson - M. Winders, Idealization of a Module, J. Commutative Algebra, 1 (2009), 3-56. | DOI | MR | Zbl

[5] N. R. Baeth - R. A. Wiegand, Factorization theory and decomposition of modules, Am. Math. Mon. 120 (2013), 3-34. | DOI | MR | Zbl

[6] S. T. CHAPMAN (ed.), Arithmetical Properties of Commutative Rings and Monoids, Lect. Notes Pure Appl. Math., vol. 241, Chapman & Hall/CRC, 2005. | DOI | MR | Zbl

[7] C. Frei - S. Frisch, Non-unique factorization of polynomials over residue class rings of the integers, Comm. Algebra, 39 (2011), 1482-1490. | DOI | MR | Zbl

[8] A. Geroldinger - F. Halter-Koch, Non-unique factorizations, vol. 278 of Pure and Applied Mathematics (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2006. Algebraic, combinatorial and analytic theory. | DOI | MR | Zbl

[9] A. Geroldinger - W. Hassler, Arithmetic of Mori domains and monoids, J. Algebra 329 (2008) 3419-3463. | DOI | MR | Zbl

[10] A. Geroldinger - W. Hassler, Local tameness of v-noetherian monoids, J. Pure Appl. Algebra, 212 (2008), 1509-1524. | DOI | MR | Zbl

[11] J. Huckaba, Commutative rings with zero divisors, Dekker, New York, 1988. | MR | Zbl

[12] A. Reinhart, On integral domains that are C-monoids, Houston J. Math., to appear. | MR | Zbl

[13] L. Salce, Transfinite self-idealization and commutative rings of triangular matrices, in Commutative algebra and its applications, 333-347, Walter de Gruyter, Berlin, 2009. | MR | Zbl