| Title:
|
On the Anderson-Badawi $\omega_{R[X]}(I[X])=\omega_R(I)$ conjecture (English) |
| Author:
|
Nasehpour, Peyman |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
52 |
| Issue:
|
2 |
| Year:
|
2016 |
| Pages:
|
71-78 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $R$. (English) |
| Keyword:
|
$n$-absorbing ideals |
| Keyword:
|
strongly $n$-absorbing ideals |
| Keyword:
|
polynomial rings |
| Keyword:
|
content algebras |
| Keyword:
|
Dedekind-Mertens content formula |
| Keyword:
|
Prüfer domains |
| Keyword:
|
Gaussian algebras |
| Keyword:
|
Gaussian rings |
| MSC:
|
13A15 |
| MSC:
|
13B02 |
| MSC:
|
13B25 |
| MSC:
|
13F05 |
| idZBL:
|
Zbl 06644059 |
| idMR:
|
MR3535629 |
| DOI:
|
10.5817/AM2016-2-71 |
| . |
| Date available:
|
2016-07-19T11:25:51Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145746 |
| . |
| Reference:
|
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