| Title:
|
The maximal regular ideal of some commutative rings (English) |
| Author:
|
Osba, Emad Abu |
| Author:
|
Henriksen, Melvin |
| Author:
|
Alkam, Osama |
| Author:
|
Smith, F. A. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
47 |
| Issue:
|
1 |
| Year:
|
2006 |
| Pages:
|
1-10 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In 1950 in volume 1 of Proc. Amer. Math. Soc., B. Brown and N. McCoy showed that every (not necessarily commutative) ring $R$ has an ideal $\frak M (R)$ consisting of elements $a$ for which there is an $x$ such that $axa=a$, and maximal with respect to this property. Considering only the case when $R$ is commutative and has an identity element, it is often not easy to determine when $\frak M (R)$ is not just the zero ideal. We determine when this happens in a number of cases: Namely when at least one of $a$ or $1-a$ has a von Neumann inverse, when $R$ is a product of local rings (e.g., when $R$ is $\Bbb Z_{n}$ or $\Bbb Z_{n}[i]$), when $R$ is a polynomial or a power series ring, and when $R$ is the ring of all real-valued continuous functions on a topological space. (English) |
| Keyword:
|
commutative rings |
| Keyword:
|
von Neumann regular rings |
| Keyword:
|
von Neumann local rings |
| Keyword:
|
Gelfand rings |
| Keyword:
|
polynomial rings |
| Keyword:
|
power series rings |
| Keyword:
|
rings of Gaussian integers (mod $n$) |
| Keyword:
|
prime and maximal ideals |
| Keyword:
|
maximal regular ideals |
| Keyword:
|
pure ideals |
| Keyword:
|
quadratic residues |
| Keyword:
|
Stone-Čech compactification |
| Keyword:
|
$C(X)$ |
| Keyword:
|
zerosets |
| Keyword:
|
cozerosets |
| Keyword:
|
$P$-spaces |
| MSC:
|
10A10 |
| MSC:
|
13A15 |
| MSC:
|
13Fxx |
| MSC:
|
16E50 |
| MSC:
|
54G10 |
| idZBL:
|
Zbl 1150.13300 |
| idMR:
|
MR2223962 |
| . |
| Date available:
|
2009-05-05T16:55:06Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119569 |
| . |
| Reference:
|
[AHA04] Abu Osba E., Henriksen M., Alkam O.: Combining local and von Neumann regular rings.Comm. Algebra 32 (2004), 2639-2653. MR 2099923 |
| Reference:
|
[AM69] Atiyah M., Macdonald J.: Introduction to Commutative Algebra.Addison-Wesley, Reading, Mass., 1969. Zbl 0238.13001, MR 0242802 |
| Reference:
|
[B81] Brewer J.: Power Series Over Commutative Rings.Marcel Dekker, New York, 1981. Zbl 0476.13015, MR 0612477 |
| Reference:
|
[BM50] Brown B., McCoy N.: The maximal regular ideal of a ring.Proc. Amer. Math. Soc. 1 (1950), 165-171. Zbl 0036.29702, MR 0034757 |
| Reference:
|
[C84] Contessa M.: On certain classes of PM rings.Comm. Algebra 12 (1984), 1447-1469. Zbl 0545.13001, MR 0744456 |
| Reference:
|
[DO71] DeMarco G., Orsatti A.: Commutative rings in which every maximal ideal is contained in a unique maximal ideal.Proc. Amer. Math. Soc. 30 (1971), 459-466. MR 0282962 |
| Reference:
|
[GJ76] Gillman L., Jerison M.: Rings of Continuous Functions.Springer, New York, 1976. Zbl 0327.46040, MR 0407579 |
| Reference:
|
[H77] Henriksen M.: Some sufficient conditions for the Jacobson radical of a commutative ring with identity to contain a prime ideal.Portugaliae Math. 36 (1977), 257-269. Zbl 0448.13002, MR 0597848 |
| Reference:
|
[L58] Leveque W.: Topics in Number Theory.Addison-Wesley, Reading, Mass., 1958. Zbl 1009.11001 |
| Reference:
|
[M74] McDonald B.R.: Finite Rings with Identity.Marcel Dekker, New York, 1974. Zbl 0294.16012, MR 0354768 |
| . |
