| Title:
|
Conditions under which $R(x)$ and $R\langle x\rangle$ are almost Q-rings (English) |
| Author:
|
Khashan, H. A. |
| Author:
|
Al-Ezeh, H. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
43 |
| Issue:
|
4 |
| Year:
|
2007 |
| Pages:
|
231-236 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
All rings considered in this paper are assumed to be commutative with identities. A ring $R$ is a $Q$-ring if every ideal of $R$ is a finite product of primary ideals. An almost $Q$-ring is a ring whose localization at every prime ideal is a $Q$-ring. In this paper, we first prove that the statements, $R$ is an almost $ZPI$-ring and $R[x]$ is an almost $Q$-ring are equivalent for any ring $R$. Then we prove that under the condition that every prime ideal of $R(x)$ is an extension of a prime ideal of $R$, the ring $R$ is a (an almost) $Q$-ring if and only if $R(x)$ is so. Finally, we justify a condition under which $R(x)$ is an almost $Q$-ring if and only if $R\left\langle x\right\rangle $ is an almost $Q$-ring. (English) |
| Keyword:
|
$Q$-rings |
| Keyword:
|
almost $Q$-rings |
| Keyword:
|
the rings $R(x)$ and $R\langle x\rangle $ |
| MSC:
|
13A15 |
| idZBL:
|
Zbl 1155.13301 |
| idMR:
|
MR2378523 |
| . |
| Date available:
|
2008-06-06T22:51:23Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/108067 |
| . |
| Reference:
|
[1] Anderson D. D., Mahaney L. A.: Commutative rings in which every ideal is a product of primary ideals.J. Algebra 106 (1987), 528–535. Zbl 0607.13004, MR 0880975 |
| Reference:
|
[2] Anderson D. D., Anderson D. F., Markanda R.: The rings $R(x)$ and $R\left\langle x\right\rangle$.J. Algebra 95 (1985), 96–115. MR 0797658 |
| Reference:
|
[3] Heinzer W., David L.: The Laskerian property in commutative rings.J. Algebra 72 (1981), 101–114. Zbl 0498.13001, MR 0634618 |
| Reference:
|
[4] Huckaba J. A.: Commutative rings with zero divisors.Marcel Dekker, INC. New York and Basel, 1988. Zbl 0637.13001, MR 0938741 |
| Reference:
|
[5] Jayaram C.: Almost Q-rings.Arch. Math. (Brno) 40 (2004), 249–257. Zbl 1112.13004, MR 2107019 |
| Reference:
|
[6] Kaplansky I.: Commutative Rings.Allyn and Bacon, Boston 1970. Zbl 0203.34601, MR 0254021 |
| Reference:
|
[7] Larsen M., McCarthy P.: Multiplicative theory of ideals.Academic Press, New York and London 1971. Zbl 0237.13002, MR 0414528 |
| Reference:
|
[8] Ohm J., Pendleton R. L.: Rings with Noetherian spectrum.Duke Math. J. 35 (1968), 631–640. Zbl 0172.32202, MR 0229627 |
| . |
