| Title:
|
Uppers to zero in $R[x]$ and almost principal ideals (English) |
| Author:
|
Borna, Keivan |
| Author:
|
Mohajer-Naser, Abolfazl |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
63 |
| Issue:
|
2 |
| Year:
|
2013 |
| Pages:
|
565-572 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $R$ be an integral domain with quotient field $K$ and $f(x)$ a polynomial of positive degree in $K[x]$. In this paper we develop a method for studying almost principal uppers to zero ideals. More precisely, we prove that uppers to zero divisorial ideals of the form $I = f(x)K[x] \cap R[x]$ are almost principal in the following two cases: – $J$, the ideal generated by the leading coefficients of $I$, satisfies $J^{-1} = R$. – $I^{-1}$ as the $R[x]$-submodule of $K(x)$ is of finite type. Furthermore we prove that for $I = f(x)K[x] \cap R[x]$ we have: – $I^{-1}\cap K[x]=(I:_{K(x)}I)$. – If there exists $p/q \in I^{-1}-K[x]$, then $(q,f)\neq 1$ in $K[x]$. If in addition $q$ is irreducible and $I$ is almost principal, then $I' = q(x)K[x] \cap R[x]$ is an almost principal upper to zero. Finally we show that a Schreier domain $R$ is a greatest common divisor domain if and only if every upper to zero in $R[x]$ contains a primitive polynomial. (English) |
| Keyword:
|
almost principal ideal |
| Keyword:
|
divisorial ideal |
| Keyword:
|
greatest common divisor domain |
| Keyword:
|
Schreier domain |
| Keyword:
|
uppers to zero |
| MSC:
|
13A05 |
| MSC:
|
13A15 |
| MSC:
|
13B25 |
| MSC:
|
13F15 |
| idZBL:
|
Zbl 06236432 |
| idMR:
|
MR3073979 |
| DOI:
|
10.1007/s10587-013-0038-9 |
| . |
| Date available:
|
2013-07-18T15:11:18Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143333 |
| . |
| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| . |
