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 locally compact...

A space $X$ is called $\mu$-compact by M. Mandelker if the intersection of all free maximal ideals of $C\left(X\right)$ coincides with the ring $C_K\left(X\right)$ of all functions in $C\left(X\right)$ with compact support. In this paper we introduce $\phi$-compact and ${\phi }^{\text{'}}$-compact spaces and we show that a space is $\mu$-compact if and only if it is both $\phi$-compact and ${\phi }^{\text{'}}$-compact. We also establish that every space $X$ admits a $\phi$-compactification and a ${\phi }^{\text{'}}$-compactification. Examples and counterexamples are given.