Isolated points and redundancy

Abstract:

We describe the isolated points of an arbitrary topological space (X, τ). If the τ-specialization pre-order on X has enough maximal éléments, then a point x ∈ X is an isolated point in (X, τ) if and only if x is both an isolated point in the subspaces of τ-kerneled points of X and in the τ-closure of {x} (a spécial case of this resuit is proved in Mehrvarz A.A., Samei K., On commutative Gelfand rings, J. Sci. Islam. Repub. Iran 10 (1999), no. 3, 193-196). This resuit is applied to an arbitrary subspace of the prime spectrum Spec(-R) of a (commutative with nonzero identity) ring R, and in particular, to the space Spec(-R) and the maximal and minimal spectrum of R. Dually, a prime idéal F of -R is an isolated point in its Zariski-kernel if and only if P is a minimal prime idéal. Finally, some aspects about the redundancy of (maximal) prime ideals in the (Jacobson) prime radical of a ring are considered, and we characterize when Spec(-R) is a scattered space.

Año de publicación:

2011

Keywords:

• Jacobson radical
• Maximal (minimal) spectrum of a ring
• Isolated point
• Scattered space
• Prime radical

Fuente:

scopus