Relative z-ideals in C(X)

AbstractFor every two ideals I⊆J in C(X), we call I a zJ-ideal if Z(f)⊆Z(g), f∈I and g∈J imply that g∈I. An ideal I is called a relative z-ideal, briefly a rez-ideal, if there exists an ideal J such that I⊊J and I is a zJ-ideal. We have shown that for any ideal J in C(X), the sum of every two zJ-ideals is a zJ-ideal if and only if X is an F-space. It is also shown that every principal ideal in C(X) is a rez-ideal if and only if X is an almost P-space and the spaces X for which the sum of every two rez-ideals is a rez-ideal are characterized. Finally for a given ideal I in C(X), the existence of greatest ideal J such that I to be a zJ-ideal and also for given two ideals I⊆J in C(X), a greatest zJ-ideal contained in I and the smallest zJ-ideal containing I are investigated

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

summary:The spaces $X$ in which every prime $z^\circ$-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 $z^\circ$-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 $z^\circ$-ideal? When is every nonregular (prime) $z$-ideal in $C(X)$ a $z^\circ$-ideal? For instance, we show that every nonregular prime ideal of $C(X)$ is a $z^\circ$-ideal if and only if $X$ is a $\partial$-space (a space in which the boundary of any zeroset is contained in a zeroset with empty interior)

On zr-Ideals of C(X)

In this paper we introduce and study a class of ideals between z-ideals and z◦-ideals (=d-ideals) namely zr-ideals. A zr-ideal is a z-ideal which is at the same time an r-ideal (an ideal I in a ring R is called an r-ideal if for each non-zerodivisor r ∈ R and each a ∈ R, ra ∈ I implies a ∈ I). In contrast to the sum of z-ideals in C(X) which is a z-ideal, the sum of zr-ideals need not be a zr-ideal. We prove that the sum of every two zr-ideals of C(X) is a zr-ideal if and only if X is a quasi F-space. In C(X) every z◦-ideal is a zr-ideal and we characterize the spacesX for which the converse is also true. We observe that X is a cozero complemented space if and only if every (prime) r-ideal in C(X) is a z-ideal and whenever every (prime) z-ideal of C(X) is an r-ideal it is equivalent to X being an almost P-space.Using these facts it turns out that the set of all r-ideals and the set of all z-ideals of C(X) coincide if and only if X is a P-space

Goldie dimension of rings of fractions of C(X)

It is observed that X is an F-space if and only if C(X) is locally a domain (i.e., C(X)p is a domain for each prime ideal P of C(X)). Consequently, X is an F-space if and only if the primary ideals of C(X) in any given maximal ideal in C(X) are comparable. Some of the properties of C(X), where X is an F-space, are extended to general reduced Bezout rings. It is observed that whenever X is an innite connected F-space, then C(X) is a natural example of a non-Noetherian ring without nontrivial idempotents which is locally a domain but not a domain. We observe that the rank of a point x ∈ βX, in case finite, coincides with the Goldie dimension of C(X)Mx and give an example to show that the Goldie dimension of C(X)Mx is not necessarily equal to the cardinality of the set of minimal prime ideals in Mx. Motivated by these facts and some other appropriate ones, we dene the rank of a point x ∈ βX to be the Goldie dimension of C(X)Mx . Finally, for each cardinal a, we show that there exists a space X and a multiplicatively closed set S in C(X) such that the Goldie dimension of S-1C(X) is α.Keywords: Goldie dimension, F-spaces, P-spaces, locally domains, reduced rings, Bezout rings, rank of a point, local cellularity, local Souslin property, at module, SV-spaces, rings of fractions

On z◦ -ideals in C(X)

An ideal I in a commutative ring R is called a z°-ideal if I consists of zero divisors and for each a ∈ I the intersection of all minimal prime ideals containing a is contained in I. We characterize topological spaces X for which z-ideals and z°-ideals coincide in , or equivalently, the sum of any two ideals consisting entirely of zero divisors consists entirely of zero divisors. Basically disconnected spaces, extremally disconnected and P-spaces are characterized in terms of z°-ideals. Finally, we construct two topological almost P-spaces X and Y which are not P-spaces and such that in every prime z°-ideal is either a minimal prime ideal or a maximal ideal and in C(Y) there exists a prime z°-ideal which is neither a minimal prime ideal nor a maximal ideal

Rings of continuous functions vanishing at infinity

summary: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 $C_\infty(X)\cong C_\infty(Y)$. It is also shown that for locally compact spaces $X$ and $Y$, $C_\infty(X)\cong C_\infty(Y)$ if and only if $X\cong Y$. 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 if every prime ideal in $C_\infty(X)$ is fixed. The existence of the smallest $\infty$-compact space in $\beta X$ containing a given space $X$ is proved. Finally some relations between topological properties of the space $X$ and algebraic properties of the ring $C_\infty(X)$ are investigated. For example we have shown that $C_\infty(X)$ is a regular ring if and only if $X$ is an $\infty$-compact $\operatorname{P}_\infty$-space