Factorization in the Self-Idealization of a PID

Gyu Whan Chang; Daniel Smertnig

Bollettino dell'Unione Matematica Italiana (2013)

  • Volume: 6, Issue: 2, page 363-377
  • ISSN: 0392-4041

Abstract

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Let D be a principal ideal domain and R ⁢ ( D ) = { ( a b 0 a ) ∣ a , b ∈ 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 ) .

How to cite

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Chang, Gyu Whan, and Smertnig, Daniel. "Factorization in the Self-Idealization of a PID." Bollettino dell'Unione Matematica Italiana 6.2 (2013): 363-377. <http://eudml.org/doc/294032>.

@article{Chang2013,
abstract = {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 $R(D)$.},
author = {Chang, Gyu Whan, Smertnig, Daniel},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {363-377},
publisher = {Unione Matematica Italiana},
title = {Factorization in the Self-Idealization of a PID},
url = {http://eudml.org/doc/294032},
volume = {6},
year = {2013},
}

TY - JOUR
AU - Chang, Gyu Whan
AU - Smertnig, Daniel
TI - Factorization in the Self-Idealization of a PID
JO - Bollettino dell'Unione Matematica Italiana
DA - 2013/6//
PB - Unione Matematica Italiana
VL - 6
IS - 2
SP - 363
EP - 377
AB - 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 $R(D)$.
LA - eng
UR - http://eudml.org/doc/294032
ER -

References

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