Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
maximal (minimal) spectrum of a ring; scattered space; isolated point; prime radical; Jacobson radical
maximal (minimal) spectrum of a ring; scattered space; isolated point; prime radical; Jacobson radical
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.
References:
[1] Heinzer W., Olberding B.: Unique irredundant intersections of completely irreducible ideals. J. Algebra 287 (2005), 432–448. DOI 10.1016/j.jalgebra.2005.03.001 | MR 2134153 | Zbl 1104.13001
[2] Henriksen M., Jerison M.: The space of minimal prime ideals of a commutative ring. Trans. Amer. Math. Soc. 115 (1965), 110–130. DOI 10.1090/S0002-9947-1965-0194880-9 | MR 0194880 | Zbl 0147.29105
[3] Hungerford T.W.: Algebra. Reprint of the 1974 original, Graduate Texts in Mathematics, 73, Springer, New York-Berlin, 1980. DOI 10.1007/978-1-4612-6101-8 | MR 0600654 | Zbl 0442.00002
[4] Mehrvarz A.A., Samei K.: On commutative Gelfand rings. J. Sci. Islam. Repub. Iran 10 (1999), no. 3, 193–196. MR 1794709 | Zbl 1061.13500
[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/
Search
Partner of
