| Title:
|
Maximal non $\lambda $-subrings (English) |
| Author:
|
Kumar, Rahul |
| Author:
|
Gaur, Atul |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
70 |
| Issue:
|
2 |
| Year:
|
2020 |
| Pages:
|
323-337 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $R$ be a commutative ring with unity. The notion of maximal non $\lambda $-subrings is introduced and studied. A ring $R$ is called a maximal non $\lambda $-subring of a ring $T$ if $R\subset T$ is not a $\lambda $-extension, and for any ring $S$ such that $R\subset S\subseteq T$, $S\subseteq T$ is a $\lambda $-extension. We show that a maximal non $\lambda $-subring $R$ of a field has at most two maximal ideals, and exactly two if $R$ is integrally closed in the given field. A determination of when the classical $D + M$ construction is a maximal non $\lambda $-domain is given. A necessary condition is given for decomposable rings to have a field which is a maximal non $\lambda $-subring. If $R$ is a maximal non $\lambda $-subring of a field $K$, where $R$ is integrally closed in $K$, then $K$ is the quotient field of $R$ and $R$ is a Prüfer domain. The equivalence of a maximal non $\lambda $-domain and a maximal non valuation subring of a field is established under some conditions. We also discuss the number of overrings, chains of overrings, and the Krull dimension of maximal non $\lambda $-subrings of a field. (English) |
| Keyword:
|
maximal non $\lambda $-subring |
| Keyword:
|
$\lambda $-extension |
| Keyword:
|
integrally closed extension |
| Keyword:
|
valuation domain |
| MSC:
|
13A18 |
| MSC:
|
13B02 |
| MSC:
|
13B22 |
| idZBL:
|
07217138 |
| idMR:
|
MR4111846 |
| DOI:
|
10.21136/CMJ.2019.0298-18 |
| . |
| Date available:
|
2020-06-17T12:30:54Z |
| Last updated:
|
2022-07-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148232 |
| . |
| Reference:
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