If $X$ is a Tychonoff space, $C\left(X\right)$ its ring of real-valued continuous functions. In this paper, we study non-essential ideals in $C\left(X\right)$. Let $a$ be a infinite cardinal, then $X$ is called $a$-Kasch (resp. $\overline{a}$-Kasch) space if given any ideal (resp. $z$-ideal) $I$ with $gen\left(I\right)<a$ then $I$ is a non-essential ideal. We show that $X$ is an ${\aleph }_0$-Kasch space if and only if $X$ is an almost $P$-space and $X$ is an ${\aleph }_1$-Kasch space if and only if $X$ is a pseudocompact and almost $P$-space. Let $C_F\left(X\right)$ denote the socle of $C\left(X\right)$. For a topological space $X$ with only...

In this short article we answer the question posed in Ghadermazi M., Karamzadeh O.A.S., Namdari M., On the functionally countable subalgebra of $C\left(X\right)$, Rend. Sem. Mat. Univ. Padova 129 (2013), 47–69. It is shown that $C_c\left(X\right)$ is isomorphic to some ring of continuous functions if and only if ${\upsilon }_{0}X$ is functionally countable. For a strongly zero-dimensional space $X$, this is equivalent to say that $X$ is functionally countable. Hence for every $P$-space it is equivalent to pseudo-${\aleph }_0$-compactness.

Let $C\left(X\right)$ be the ring of real continuous functions on a completely regular Hausdorff space. In this paper an almost discrete space is determined by the algebraic structure of $C\left(X\right)$. The intersection of essential weak ideal in $C\left(X\right)$ is also studied.

A nonzero $R$-module $M$ is atomic if for each two nonzero elements $a,b$ in $M$, both cyclic submodules $Ra$ and $Rb$ have nonzero isomorphic submodules. In this article it is shown that for an infinite $P$-space $X$, the factor rings $C(X,\mathbb{Z})/C_F(X,\mathbb{Z})$ and $C_c\left(X\right)/C_F\left(X\right)$ have no atomic ideals. This fact generalizes a result published in paper by A. Mozaffarikhah, E. Momtahan, A. R. Olfati and S. Safaeeyan (2020), which says that for an infinite set $X$, the factor ring ${\mathbb{Z}}^X/{\mathbb{Z}}^{\left(X\right)}$ has no atomic ideal. Another result is that for each infinite $P$-space $X$, the...

The spaces $X$ in which every prime $z^{\circ }$-ideal of $C\left(X\right)$ 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 $C\left(X\right)$ is either minimal or maximal. We will also answer the following questions: When is every nonregular prime ideal in $C\left(X\right)$ a $z^{\circ }$-ideal? When is every nonregular (prime) $z$-ideal in $C\left(X\right)$ a $z^{\circ }$-ideal? For...

We study the geometrical properties of a unit vector field on a Riemannian 2-manifold, considering the field as a local imbedding of the manifold into its tangent sphere bundle with the Sasaki metric. For the case of constant curvature $K$, we give a description of the totally geodesic unit vector fields for $K=0$ and $K=1$ and prove a non-existence result for $K\ne 0,1$. We also found a family ${\xi }_{\omega }$ of vector fields on the hyperbolic 2-plane $L^2$ of curvature $-c^2$ which generate foliations on $T_1{L}^2$ with leaves of constant intrinsic...

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,...