Isolated points and redundancy

P. Alirio J. Peña; Jorge E. Vielma

Displaying similar documents to “Isolated points and redundancy”

Pseudo-valuation rings. II

David F. Anderson, Ayman Badawi, David E. Dobbs (2000)

Bollettino dell'Unione Matematica Italiana

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Viene data una condizione sufficiente affinchè un sopra-anello di un anello di pseudo-valutazione (PVR) sia ancora un PVR. Da ciò segue che se $(R, M)$ è un PVR, allora ogni sopra-anello di $R$ è un PVR se (e soltanto se) $R [u]$ è quasi-locale per ciascun elemento $u$ di $(M : M)$ . Vari risultati sono dimostrati per un ideale primo di un anello commutativo arbitrario $R$ , avente $Z (R)$ come insieme di zero-divisori. Per esempio, se $P$ è un primo «forte» di $R$ e contiene un elemento non-zero divisore di $R$ , allora $(P : P)$ è...

A primrose path from Krull to Zorn

Marcel Erné (1995)

Commentationes Mathematicae Universitatis Carolinae

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Given a set $X$ of “indeterminates” and a field $F$ , an ideal in the polynomial ring $R = F [X]$ is called conservative if it contains with any polynomial all of its monomials. The map $S \mapsto R S$ yields an isomorphism between the power set $P (X)$ and the complete lattice of all conservative prime ideals of $R$ . Moreover, the members of any system $S \subseteq P (X)$ of finite character are in one-to-one correspondence with the conservative prime ideals contained in $P_{S} = ⋃ {R S : S \in S}$ , and the maximal members of $S$ correspond to the maximal ideals contained...

Rings of continuous functions vanishing at infinity

Ali Rezaei Aliabad, F. Azarpanah, M. Namdari (2004)

Commentationes Mathematicae Universitatis Carolinae

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We prove that a Hausdorff space $X$ is locally compact if and only if its topology coincides with the weak topology induced by $C_{\infty} (X)$ . It is shown that for a Hausdorff space $X$ , there exists a locally compact Hausdorff space $Y$ such that $C_{\infty} (X) ≅ C_{\infty} (Y)$ . It is also shown that for locally compact spaces $X$ and $Y$ , $C_{\infty} (X) ≅ C_{\infty} (Y)$ if and only if $X ≅ Y$ . Prime ideals in $C_{\infty} (X)$ are uniquely represented by a class of prime ideals in $C^{*} (X)$ . $\infty$ -compact spaces are introduced and it turns out that a locally compact space $X$ is $\infty$ -compact if and only...

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 $I \subseteq 𝒵 (ℛ)$ and $I$ is contained in no minimal prime ideal. We denote by $R_{K} (ℳ)$ , the set of all $a \in R$ for which $\bar{D (a)} = \bar{ℳ ∖ V (a)}$ 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...