Let ${\left\{X_{\alpha }\right\}}_{\alpha \in \Lambda }$ be a family of topological spaces and $x_{\alpha }\in X_{\alpha }$, for every $\alpha \in \Lambda$. Suppose $X$ is the quotient space of the disjoint union of $X_{\alpha }$’s by identifying $x_{\alpha }$’s as one point $\sigma$. We try to characterize ideals of $C\left(X\right)$ according to the same ideals of $C\left(X_{\alpha }\right)$’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)...

Let $I$ be a semi-prime ideal. Then $P_{\circ }\in Min\left(I\right)$ is called irredundant with respect to $I$ if $I\ne {\bigcap }_{P_{\circ }\ne P\in Min\left(I\right)}P$. If $I$ is the intersection of all irredundant ideals with respect to $I$, it is called a fixed-place ideal. If there are no irredundant ideals with respect to $I$, it is called an anti fixed-place ideal. We show that each semi-prime ideal has a unique representation as an intersection of a fixed-place ideal and an anti fixed-place ideal. We say the point $p\in \beta X$ is a fixed-place point if $O^p\left(X\right)$ is a fixed-place ideal. In...

Let $𝒵\left(ℛ\right)$ 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 $sd$-ideal if $I\subseteq 𝒵\left(ℛ\right)$ and $I$ is contained in no minimal prime ideal. We denote by $R_K\left(ℳ\right)$, the set of all $a\in R$ for which $\overline{D\left(a\right)}=\overline{ℳ\setminus V\left(a\right)}$ is compact. We show that $R$ has property $\left(A\right)$ and $ℳ$ is compact if and only if $R$ has no $sd$-ideal. It is proved that $R_K\left(ℳ\right)$ is an essential ideal (resp., $sd$-ideal) if and only if $ℳ$ is an almost...

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