Displaying similar documents to “Intersections of minimal prime ideals in the rings of continuous functions”

Intersections of essential minimal prime ideals

A. Taherifar (2014)

Commentationes Mathematicae Universitatis Carolinae

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Let 𝒵 ( ℛ ) be the set of zero divisor elements of a commutative ring R with identity and ℳ be the space of minimal prime ideals of R with Zariski topology. An ideal I of R is called strongly dense ideal or briefly s d -ideal if I ⊆ 𝒵 ( ℛ ) and I is contained in no minimal prime ideal. We denote by R K ( ℳ ) , the set of all a ∈ R for which D ( a ) ¯ = ℳ ∖ V ( a ) ¯ is compact. We show that R has property ( A ) and ℳ is compact if and only if R has no s d -ideal. It is proved that R K ( ℳ ) is an essential ideal (resp., s d -ideal) if and only if ℳ is an almost...

Pasting topological spaces at one point

Ali Rezaei Aliabad (2006)

Czechoslovak Mathematical Journal

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Let { X α } α ∈ Λ be a family of topological spaces and x α ∈ X α , for every α ∈ Λ . Suppose X is the quotient space of the disjoint union of X α ’s by identifying x α ’s as one point σ . We try to characterize ideals of C ( X ) according to the same ideals of C ( X α ) ’s. In addition we generalize the concept of rank of a point, see [9], and then answer the following two algebraic questions. Let m be an infinite cardinal. (1) Is there any ring R and I an ideal in R such that I is an irreducible intersection of m prime ideals? (2)...

Isolated points and redundancy

P. Alirio J. Peña, Jorge E. Vielma (2011)

Commentationes Mathematicae Universitatis Carolinae

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We describe the isolated points of an arbitrary topological space ( X , τ ) . If the τ -specialization pre-order on X has enough maximal elements, 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 special case of this result is proved in Mehrvarz A.A., Samei K., , J. Sci. Islam. Repub. Iran (1999), no. 3, 193–196). This result is applied to an arbitrary subspace of the prime spectrum Spec ( R ) of...

On nonregular ideals and z ∘ -ideals in C ( X )

F. Azarpanah, M. Karavan (2005)

Czechoslovak Mathematical Journal

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The spaces X in which every prime z ∘ -ideal of C ( X ) is either minimal or maximal are characterized. By this characterization, it turns out that for a large class of topological spaces X , such as metric spaces, basically disconnected spaces and one-point compactifications of discrete spaces, every prime z ∘ -ideal in C ( X ) is either minimal or maximal. We will also answer the following questions: When is every nonregular prime ideal in C ( X ) a z ∘ -ideal? When is every nonregular (prime) z -ideal in C ( X ) a...