Rostam Mohamadian | Searcharxiv

arXiv2023

Countable strongly annihilated ideals in commutative rings

In this paper we introduce and study the concept of countable strongly annihilated ideal in commutative rings, in particular in rings of continuous functions. We show that a maximal ideal in $C(X)$ is countable strongly annihilated if and only if it is a real maximal $z^\circ$-ideal. It turns out that $X$ is an almost $P$-space if and only if countable strongly annihilated ideals and strongly divisible $z$-ideals coincide if and only if every $M_x$ is a countable strongly annihilated ideal, for any $x\in X$. We observe that an almost $P$-space $X$ is Lindelof if and only if every countable strongly annihilated ideal is fixed. We give a negative answer to a question raised by Gilmer and McAdam.

math.AC↗

arXiv2022

Locally closed sets and submaximal spaces

A topological space $X$ is called submaximal if every dense subset of $X$ is open. In this paper, we show that if $βX$, the Stone-Čech compactification of $X$, is a submaximal space, then $X$ is a compact space and hence $βX=X$. We also prove that if $\upsilon X$, the Hewitt realcompactification of $X$, is submaximal and first countable and $X$ is without isolated point, then $X$ is realcompact and hence $\upsilon X=X$. We observe that every submaximal Hausdorff space is pseudo-finite. It turns out that if $\upsilon X$ is a submaximal space, then $X$ is a pseudo-finite $μ$-compact space. An example is given which shows that $X$ may be submaximal but $\upsilon X$ may not be submaximal. Given a topological space $(X,{\mathcal T})$, the collection of all locally closed subsets of $X$ forms a base for a topology on $X$ which is denotes by ${\mathcal T_l}$. We study some topological properties between $(X,{\mathcal T})$ and $(X,{\mathcal T_l})$, such as we show that a) $(X,{\mathcal T_l})$ is discrete if and only if $(X,{\mathcal T})$ is a $T_D$-space; b) $(X,{\mathcal T})$ is a locally indiscrete space if and only if ${\mathcal T}={\mathcal T_l}$; c) $(X,{\mathcal T})$ is indiscrete space if and only if $(X,{\mathcal T_l})$ is connected. We see that, in locally indiscrete spaces, the concepts of $T_0$, $T_D$, $T_\frac{1}{2}$, $T_1$, submaximal and discrete coincide. Finally, we prove that every clopen subspace of an lc-compact space is lc-compact.

math.GN↗