| Title:
|
The Bordalo order on a commutative ring (English) |
| Author:
|
Henriksen, Melvin |
| Author:
|
Smith, F. A. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
40 |
| Issue:
|
3 |
| Year:
|
1999 |
| Pages:
|
429-440 |
| . |
| Category:
|
math |
| . |
| Summary:
|
If $R$ is a commutative ring with identity and $\leq$ is defined by letting $a\leq b$ mean $ab=a$ or $a=b$, then $(R,\leq)$ is a partially ordered ring. Necessary and sufficient conditions on $R$ are given for $(R,\leq)$ to be a lattice, and conditions are given for it to be modular or distributive. The results are applied to the rings $Z_{n}$ of integers mod $n$ for $n\geq2$. In particular, if $R$ is reduced, then $(R,\leq)$ is a lattice iff $R$ is a weak Baer ring, and $(R,\leq)$ is a distributive lattice iff $R$ is a Boolean ring, $Z_{3},Z_{4}$, $Z_{2}[x]/x^{2}Z_{2}[x]$, or a four element field. (English) |
| Keyword:
|
commutative ring |
| Keyword:
|
reduced ring |
| Keyword:
|
integral domain |
| Keyword:
|
field |
| Keyword:
|
connected ring |
| Keyword:
|
\linebreak Boolean ring |
| Keyword:
|
weak Baer Ring |
| Keyword:
|
regular element |
| Keyword:
|
annihilator |
| Keyword:
|
nilpotents |
| Keyword:
|
idempotents |
| Keyword:
|
cover |
| Keyword:
|
partial order |
| Keyword:
|
incomparable elements |
| Keyword:
|
lattice |
| Keyword:
|
modular lattice |
| Keyword:
|
distributive lattice |
| MSC:
|
03G10 |
| MSC:
|
06A06 |
| MSC:
|
06F25 |
| MSC:
|
11A07 |
| MSC:
|
13A99 |
| idZBL:
|
Zbl 1011.06019 |
| idMR:
|
MR1732492 |
| . |
| Date available:
|
2009-01-08T18:53:45Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119099 |
| . |
| Reference:
|
[Be] Berberian S.: Baer$^*$-rings.Springer-Verlag, New York, 1972. Zbl 0679.16011, MR 0429975 |
| Reference:
|
[Bo] Bordalo G.: Naturally ordered commutative rings.preprint. |
| Reference:
|
[ES] Speed T., Evans M.: A note on commutative Baer rings.J. Austral. Math. Soc. 13 (1971), 1-6. MR 0294318 |
| Reference:
|
[J] Jacobson N.: Basic Algebra I.W.H. Freeman and Co., San Francisco, 1974. Zbl 0557.16001, MR 0356989 |
| Reference:
|
[K] Kist J.: Minimal prime ideals in commutative semigroups.Proc. London Math. Soc. 13 (1963), 31-50. Zbl 0108.04004, MR 0143837 |
| Reference:
|
[Sp1] Speed T.: A note on commutative Baer rings I.J. Austral. Math. Soc. 14 (1972), 257-263. MR 0318120 |
| Reference:
|
[Sp2] Speed T.: A note on commutative Baer rings II.ibid. 15 (1973), 15-21. MR 0330140 |
| . |
