On the Anderson-Badawi ${\omega }_{R\left[X\right]}\left(I\left[X\right]\right)={\omega }_R\left(I\right)$ conjecture

@article{Nasehpour2016,
abstract = {Let $R$ be a commutative ring with an identity different from zero and $n$ be a positive integer. Anderson and Badawi, in their paper on $n$-absorbing ideals, define a proper ideal $I$ of a commutative ring $R$ to be an $n$-absorbing ideal of $R$, if whenever $x_1 \dots x_\{n+1\} \in I$ for $x_1, \ldots , x_\{n+1\} \in R$, then there are $n$ of the $x_i$’s whose product is in $I$ and conjecture that $\omega _\{R[X]\}(I[X])=\omega _R(I)$ for any ideal $I$ of an arbitrary ring $R$, where $\omega _R(I)= \min \lbrace n\colon I \text\{is\} \text\{an\} n\text\{-absorbing\} \text\{ideal\} \text\{of\} R\rbrace $. In the present paper, we use content formula techniques to prove that their conjecture is true, if one of the following conditions hold: The ring $R$ is a Prüfer domain. The ring $R$ is a Gaussian ring such that its additive group is torsion-free. The additive group of the ring $R$ is torsion-free and $I$ is a radical ideal of $R$.},
author = {Nasehpour, Peyman},
journal = {Archivum Mathematicum},
keywords = {$n$-absorbing ideals; strongly $n$-absorbing ideals; polynomial rings; content algebras; Dedekind-Mertens content formula; Prüfer domains; Gaussian algebras; Gaussian rings},
language = {eng},
number = {2},
pages = {71-78},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On the Anderson-Badawi $\omega _\{R[X]\}(I[X])=\omega _R(I)$ conjecture},
url = {http://eudml.org/doc/281547},
volume = {052},
year = {2016},
}

TY - JOUR
AU - Nasehpour, Peyman
TI - On the Anderson-Badawi $\omega _{R[X]}(I[X])=\omega _R(I)$ conjecture
JO - Archivum Mathematicum
PY - 2016
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 052
IS - 2
SP - 71
EP - 78
AB - Let $R$ be a commutative ring with an identity different from zero and $n$ be a positive integer. Anderson and Badawi, in their paper on $n$-absorbing ideals, define a proper ideal $I$ of a commutative ring $R$ to be an $n$-absorbing ideal of $R$, if whenever $x_1 \dots x_{n+1} \in I$ for $x_1, \ldots , x_{n+1} \in R$, then there are $n$ of the $x_i$’s whose product is in $I$ and conjecture that $\omega _{R[X]}(I[X])=\omega _R(I)$ for any ideal $I$ of an arbitrary ring $R$, where $\omega _R(I)= \min \lbrace n\colon I \text{is} \text{an} n\text{-absorbing} \text{ideal} \text{of} R\rbrace $. In the present paper, we use content formula techniques to prove that their conjecture is true, if one of the following conditions hold: The ring $R$ is a Prüfer domain. The ring $R$ is a Gaussian ring such that its additive group is torsion-free. The additive group of the ring $R$ is torsion-free and $I$ is a radical ideal of $R$.
LA - eng
KW - $n$-absorbing ideals; strongly $n$-absorbing ideals; polynomial rings; content algebras; Dedekind-Mertens content formula; Prüfer domains; Gaussian algebras; Gaussian rings
UR - http://eudml.org/doc/281547
ER -