Valuation Ideal Factorization Domains
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866918259822952448 |
|---|---|
| author | Chang, Gyu Whan Reinhart, Andreas |
| author_facet | Chang, Gyu Whan Reinhart, Andreas |
| contents | An integral domain $D$ is a {\em valuation ideal factorization domain} (VIFD) if each nonzero principal ideal of $D$ can be written as a finite product of valuation ideals. Clearly, $π$-domains are VIFDs. We study the ring-theoretic properties of VIFDs and the $*$-operation analogs of VIFDs. Among them, we show that if $D$ is treed (resp., $*$-treed), then $D$ is a VIFD (resp., $*$-VIFD) if and only if $D$ is an ${\rm h}$-local Prüfer domain (resp., a $*$-${\rm h}$-local P$*$MD) if and only if every nonzero prime ideal of $D$ contains an invertible (resp., a $*$-invertible) valuation ideal. We also study integral domains $D$ such that for each nonzero nonunit $a\in D$, there is a positive integer $n$ such that $a^n$ can be written as a finite product of valuation elements. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16471 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Valuation Ideal Factorization Domains Chang, Gyu Whan Reinhart, Andreas Commutative Algebra 13A15, 13F05, 13G05 An integral domain $D$ is a {\em valuation ideal factorization domain} (VIFD) if each nonzero principal ideal of $D$ can be written as a finite product of valuation ideals. Clearly, $π$-domains are VIFDs. We study the ring-theoretic properties of VIFDs and the $*$-operation analogs of VIFDs. Among them, we show that if $D$ is treed (resp., $*$-treed), then $D$ is a VIFD (resp., $*$-VIFD) if and only if $D$ is an ${\rm h}$-local Prüfer domain (resp., a $*$-${\rm h}$-local P$*$MD) if and only if every nonzero prime ideal of $D$ contains an invertible (resp., a $*$-invertible) valuation ideal. We also study integral domains $D$ such that for each nonzero nonunit $a\in D$, there is a positive integer $n$ such that $a^n$ can be written as a finite product of valuation elements. |
| title | Valuation Ideal Factorization Domains |
| topic | Commutative Algebra 13A15, 13F05, 13G05 |
| url | https://arxiv.org/abs/2410.16471 |