Isolated points and redundancy

Alirio J. Peña, P., and Vielma, Jorge E.. "Isolated points and redundancy." Commentationes Mathematicae Universitatis Carolinae 52.1 (2011): 145-152. <http://eudml.org/doc/247084>.

@article{AlirioJ2011,
abstract = {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 $\lbrace x\rbrace $ (a special case of this result is proved in Mehrvarz A.A., Samei K., On commutative Gelfand rings, J. Sci. Islam. Repub. Iran 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.},
author = {Alirio J. Peña, P., Vielma, Jorge E.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
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},
language = {eng},
number = {1},
pages = {145-152},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Isolated points and redundancy},
url = {http://eudml.org/doc/247084},
volume = {52},
year = {2011},
}

TY - JOUR
AU - Alirio J. Peña, P.
AU - Vielma, Jorge E.
TI - Isolated points and redundancy
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2011
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 52
IS - 1
SP - 145
EP - 152
AB - 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 $\lbrace x\rbrace $ (a special case of this result is proved in Mehrvarz A.A., Samei K., On commutative Gelfand rings, J. Sci. Islam. Repub. Iran 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.
LA - eng
KW - 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
UR - http://eudml.org/doc/247084
ER -