# Star operations in extensions of integral domains

• [1] Department of Mathematics, University of Tennessee Knoxville, TN 37996, USA
• [2] Department of Mathematics, Faculté des Sciences et Techniques P.O. Box 523, Beni Mellal, Morocco
• [3] 57 Colgate Street, Pocatello, ID 83201, USA
• Volume: 2, Issue: 2, page 87-89
• ISSN: 2105-0597

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## Abstract

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An extension $D\subseteq R$ of integral domains is strongly$t$-compatible (resp., $t$-compatible) if ${\left(IR\right)}^{-1}={\left({I}^{-1}R\right)}_{v}$ (resp., ${\left(IR\right)}_{v}={\left({I}_{v}R\right)}_{v}\right)$ for every nonzero finitely generated fractional ideal $I$ of $D$. We show that strongly $t$-compatible implies $t$-compatible and give examples to show that the converse does not hold. We also indicate situations where strong $t$-compatibility and its variants show up naturally. In addition, we study integral domains $D$ such that $D\subseteq R$ is strongly $t$-compatible (resp., $t$-compatible) for every overring $R$ of $D$.

## How to cite

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Anderson, David F., El Baghdadi, Said, and Zafrullah, Muhammad. "Star operations in extensions of integral domains." Actes des rencontres du CIRM 2.2 (2010): 87-89. <http://eudml.org/doc/196282>.

@article{Anderson2010,
abstract = {An extension $D \subseteq R$ of integral domains is strongly$t$-compatible (resp., $t$-compatible) if $(IR)^\{-1\} = (I^\{-1\}R)_\{v\}$ (resp., $(IR)_\{v\} = (I_\{v\}R)_\{v\})$ for every nonzero finitely generated fractional ideal $I$ of $D$. We show that strongly $t$-compatible implies $t$-compatible and give examples to show that the converse does not hold. We also indicate situations where strong $t$-compatibility and its variants show up naturally. In addition, we study integral domains $D$ such that $D \subseteq R$ is strongly $t$-compatible (resp., $t$-compatible) for every overring $R$ of $D$.},
affiliation = {Department of Mathematics, University of Tennessee Knoxville, TN 37996, USA; Department of Mathematics, Faculté des Sciences et Techniques P.O. Box 523, Beni Mellal, Morocco; 57 Colgate Street, Pocatello, ID 83201, USA},
journal = {Actes des rencontres du CIRM},
keywords = {Star operation; $t$-linked; $t$-compatible; strongly $t$-compatible; domain extensions; Prüfer domain},
language = {eng},
number = {2},
pages = {87-89},
publisher = {CIRM},
title = {Star operations in extensions of integral domains},
url = {http://eudml.org/doc/196282},
volume = {2},
year = {2010},
}

TY - JOUR
AU - Anderson, David F.
TI - Star operations in extensions of integral domains
JO - Actes des rencontres du CIRM
PY - 2010
PB - CIRM
VL - 2
IS - 2
SP - 87
EP - 89
AB - An extension $D \subseteq R$ of integral domains is strongly$t$-compatible (resp., $t$-compatible) if $(IR)^{-1} = (I^{-1}R)_{v}$ (resp., $(IR)_{v} = (I_{v}R)_{v})$ for every nonzero finitely generated fractional ideal $I$ of $D$. We show that strongly $t$-compatible implies $t$-compatible and give examples to show that the converse does not hold. We also indicate situations where strong $t$-compatibility and its variants show up naturally. In addition, we study integral domains $D$ such that $D \subseteq R$ is strongly $t$-compatible (resp., $t$-compatible) for every overring $R$ of $D$.
LA - eng
KW - Star operation; $t$-linked; $t$-compatible; strongly $t$-compatible; domain extensions; Prüfer domain
UR - http://eudml.org/doc/196282
ER -

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