| Title:
|
On atomic ideals in some factor rings of $C(X,\Bbb Z)$ (English) |
| Author:
|
Olfati, Alireza |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
62 |
| Issue:
|
2 |
| Year:
|
2021 |
| Pages:
|
259-263 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A nonzero $R$-module $M$ is atomic if for each two nonzero elements $a, b$ in $M$, both cyclic submodules $Ra$ and $Rb$ have nonzero isomorphic submodules. In this article it is shown that for an infinite $P$-space $X$, the factor rings $C(X,\Bbb{Z})/C_F(X,\Bbb{Z})$ and $C_c(X)/C_F(X)$ have no atomic ideals. This fact generalizes a result published in paper by A. Mozaffarikhah, E. Momtahan, A. R. Olfati and S. Safaeeyan (2020), which says that for an infinite set $X$, the factor ring $\Bbb{Z}^X/ \Bbb{Z}^{(X)}$ has no atomic ideal. Another result is that for each infinite $P$-space $X$, the socle of the factor ring $C_c(X)/C_F(X)$ is always equal to zero. Also, zero-dimensional spaces $X$ are characterized for which $C^F(X,\Bbb{Z})/C_F(X,\Bbb{Z})$ have atomic ideals. (English) |
| Keyword:
|
$P$-space |
| Keyword:
|
rings of integer-valued continuous functions |
| Keyword:
|
functionally countable subalgebra |
| Keyword:
|
atomic ideal |
| Keyword:
|
socle |
| MSC:
|
54C40 |
| idZBL:
|
Zbl 07396222 |
| idMR:
|
MR4303551 |
| DOI:
|
10.14712/1213-7243.2021.013 |
| . |
| Date available:
|
2021-07-28T08:41:36Z |
| Last updated:
|
2023-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149015 |
| . |
| 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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| Reference:
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| Reference:
|
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| . |
