| Title:
|
On commutative rings whose prime ideals are direct sums of cyclics (English) |
| Author:
|
Behboodi, M. |
| Author:
|
Moradzadeh-Dehkordi, A. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
48 |
| Issue:
|
4 |
| Year:
|
2012 |
| Pages:
|
291-299 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we study commutative rings $R$ whose prime ideals are direct sums of cyclic modules. In the case $R$ is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring $(R, \mathcal{M})$, the following statements are equivalent: (1) Every prime ideal of $R$ is a direct sum of cyclic $R$-modules; (2) ${\mathcal{M}}=\bigoplus _{\lambda \in \Lambda }Rw_{\lambda }$ where $\Lambda $ is an index set and $R/{\operatorname{Ann}}(w_{\lambda })$ is a principal ideal ring for each $\lambda \in \Lambda $; (3) Every prime ideal of $R$ is a direct sum of at most $|\Lambda |$ cyclic $R$-modules where $\Lambda $ is an index set and ${\mathcal{M}}=\bigoplus _{\lambda \in \Lambda }Rw_{\lambda }$; and (4) Every prime ideal of $R$ is a summand of a direct sum of cyclic $R$-modules. Also, we establish a theorem which state that, to check whether every prime ideal in a Noetherian local ring $(R, \mathcal{M})$ is a direct sum of (at most $n$) principal ideals, it suffices to test only the maximal ideal $\mathcal{M}$. (English) |
| Keyword:
|
prime ideals |
| Keyword:
|
cyclic modules |
| Keyword:
|
local rings |
| Keyword:
|
principal ideal rings |
| MSC:
|
13C05 |
| MSC:
|
13E05 |
| MSC:
|
13E10 |
| MSC:
|
13F10 |
| MSC:
|
13H99 |
| idMR:
|
MR3007611 |
| DOI:
|
10.5817/AM2012-4-291 |
| . |
| Date available:
|
2012-12-17T13:52:53Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143103 |
| . |
| Reference:
|
[1] Anderson, D. D.: A note on minimal prime ideals.Proc. Amer. Math. Soc. 122 (1994), 13–14. Zbl 0841.13001, MR 1191864, 10.1090/S0002-9939-1994-1191864-2 |
| Reference:
|
[2] Behboodi, M., Ghorbani, A., Moradzadeh–Dehkordi, A.: Commutative Noetherian local rings whose ideals are direct sums of cyclic modules.J. Algebra 345 (2011), 257–265. Zbl 1244.13008, MR 2842065, 10.1016/j.jalgebra.2011.08.017 |
| Reference:
|
[3] Behboodi, M., Ghorbani, A., Moradzadeh–Dehkordi, A., Shojaee, S. H.: On left Köthe rings and a generalization of the Köthe–Cohen–Kaplansky theorem.Proc. Amer. Math. Soc. (to appear). MR 2530766 |
| Reference:
|
[4] Behboodi, M., Shojaee, S. H.: Commutative local rings whose ideals are direct sums of cyclic modules.submitted. |
| Reference:
|
[5] Cohen, I. S.: Commutative rings with restricted minimum condition.Duke Math. J. 17 (1950), 27–42. Zbl 0041.36408, MR 0033276, 10.1215/S0012-7094-50-01704-2 |
| Reference:
|
[6] Cohen, I. S., Kaplansky, I.: Rings for which every module is a direct sum of cyclic modules.Math. Z. 54 (1951), 97–101. Zbl 0043.26702, MR 0043073, 10.1007/BF01179851 |
| Reference:
|
[7] Kaplansky, I.: Elementary divisors and modules.Trans. Amer. Math. Soc. 66 (1949), 464–491. Zbl 0036.01903, MR 0031470, 10.1090/S0002-9947-1949-0031470-3 |
| Reference:
|
[8] Köthe, G.: Verallgemeinerte abelsche gruppen mit hyperkomplexen operatorenring.Math. Z. 39 (1935), 31–44. MR 1545487, 10.1007/BF01201343 |
| Reference:
|
[9] Warfield, R. B. Jr., : A Krull–Schmidt theorem for infinite sums of modules.Proc. Amer. Math. Soc. 22 (1969), 460–465. Zbl 0176.31401, MR 0242886, 10.1090/S0002-9939-1969-0242886-2 |
| . |
