| Title:
|
Isolated points and redundancy (English) |
| Author:
|
Alirio J. Peña, P. |
| Author:
|
Vielma, Jorge |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
52 |
| Issue:
|
1 |
| Year:
|
2011 |
| Pages:
|
145-152 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We describe the isolated points of an arbitrary topological space $(X,\tau)$. If the $\tau$-specialization pre-order on $X$ has enough maximal elements, then a point $x\in X$ is an isolated point in $(X,\tau)$ if and only if $x$ is both an isolated point in the subspaces of $\tau$-kerneled points of $X$ and in the $\tau$-closure of $\{x\}$ (a special case of this result is proved in Mehrvarz A.A., Samei K., {\it On commutative Gelfand rings\/}, J. Sci. Islam. Repub. Iran {\bf 10} (1999), no. 3, 193--196). This result is applied to an arbitrary subspace of the prime spectrum $\operatorname{Spec}(R)$ of a (commutative with nonzero identity) ring $R$, and in particular, to the space $\operatorname{Spec}(R)$ and the maximal and minimal spectrum of $R$. Dually, a prime ideal $P$ of $R$ is an isolated point in its Zariski-kernel if and only if $P$ is a minimal prime ideal. Finally, some aspects about the redundancy of (maximal) prime ideals in the (Jacobson) prime radical of a ring are considered, and we characterize when $\operatorname{Spec} (R)$ is a scattered space. (English) |
| Keyword:
|
maximal (minimal) spectrum of a ring |
| Keyword:
|
scattered space |
| Keyword:
|
isolated point |
| Keyword:
|
prime radical |
| Keyword:
|
Jacobson radical |
| MSC:
|
13C05 |
| MSC:
|
54F65 |
| idZBL:
|
Zbl 1240.54106 |
| idMR:
|
MR2828365 |
| . |
| Date available:
|
2011-03-08T17:44:19Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141434 |
| . |
| Reference:
|
[1] Heinzer W., Olberding B.: Unique irredundant intersections of completely irreducible ideals.J. Algebra 287 (2005), 432–448. Zbl 1104.13001, MR 2134153, 10.1016/j.jalgebra.2005.03.001 |
| Reference:
|
[2] Henriksen M., Jerison M.: The space of minimal prime ideals of a commutative ring.Trans. Amer. Math. Soc. 115 (1965), 110–130. Zbl 0147.29105, MR 0194880, 10.1090/S0002-9947-1965-0194880-9 |
| Reference:
|
[3] Hungerford T.W.: Algebra.Reprint of the 1974 original, Graduate Texts in Mathematics, 73, Springer, New York-Berlin, 1980. Zbl 0442.00002, MR 0600654, 10.1007/978-1-4612-6101-8 |
| Reference:
|
[4] Mehrvarz A.A., Samei K.: On commutative Gelfand rings.J. Sci. Islam. Repub. Iran 10 (1999), no. 3, 193–196. Zbl 1061.13500, MR 1794709 |
| Reference:
|
[5] Peña A.J., Ruza L.M., Vielma J.: Separation axioms and the prime spectrum of commutative semirings.Notas de Matemática, Vol. 5 (2), No. 284, 2009, pp. 66–82; http://www.saber.ula.ve/notasdematematica/. |
| . |
