Isolated points and redundancy

Alirio J. Peña, P. ; Vielma, Jorge

Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 1, pp. 145-152.

Résumé

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.

Détail

Classification : 13C05, 54F65
Keywords: maximal (minimal) spectrum of a ring; scattered space; isolated point; prime radical; Jacobson radical