Search

For a measurable space $(X,\mathcal{A})$, let $\mathcal{M}^+(X,\mathcal{A})$ be the commutative semiring of non-negative real-valued measurable functions with pointwise addition and pointwise multiplication. We show that there is a lattice isomorphism between the ideal lattice of $\mathcal{M}^+(X,\mathcal{A})$ and the ideal lattice of its ring of differences $\mathcal{M}(X,\mathcal{A})$. Moreover, we infer that each ideal of $\mathcal{M}^+(X,\mathcal{A})$ is a semiring $z$-ideal. We investigate the duality between cancellative congruences on $\mathcal{M}^{+}(X,\mathcal{A})$ and $Z_{\mathcal{A}}$-filters on $X$. We observe that for $\sigma$-algebras, compactness and pseudocompactness coincide, and we provide a new characterization for compact measurable spaces via algebraic properties of $\mathcal{M}^+(X,\mathcal{A})$. It is shown that the space of (real) maximal congruences on $\mathcal{M}^+(X,\mathcal{A})$ is homeomorphic to the space of (real) maximal ideals of the $\mathcal{M}(X,\mathcal{A})$. We solve the isomorphism problem for the semirings of the form $\mathcal{M}^+(X,\mathcal{A})$ for compact and realcompact measurable spaces.

Our aim is to study certain algebraic properties of the ring $C(X)_\mathcal{P}$ of real-valued functions on $X$ whose closure of discontinuity set is in an ideal of closed sets. We characterize $\mathcal{P}P$-spaces using $z$-ideals and essential ideals of $C(X)_\mathcal{P}$ and also almost $\mathcal{P}P$-spaces using $z^0$-ideals of $C(X)_\mathcal{P}$ and a topology finer than the original topology on $X$. We deduce that each maximal ideal of $C(X)_F$ \cite{GGT2018} (resp. $T'(X)$ \cite{A2010}) is a $z^0$-ideal. We establish that the notions of clean ring, weakly clean ring, semiclean ring, almost clean ring and exchange ring coincide in the ring $C(X)_\mathcal{P}$. End of this paper, we also characterize $\mathcal{P}P$-spaces and almost $\mathcal{P}P$-spaces using certain ideals having depth zero. We exhibit a condition on $\mathcal{P}$ under which prime and essential ideals of $C(X)_\mathcal{P}$ have depth zero.

Consider the ring $C_c(X)_F$ of real valued functions which are discontinuous on a finite set with countable range. We discuss $(\mathcal{Z}_c)_F$-filters on $X$ and $(\mathcal{Z}_c)_F$-ideals of $C_c(X)_F$. We establish an analogous version of Gelfand-Kolmogoroff theorem in our setting. We prove some equivalent conditions when $C_c(X)_F$ is a Baer-ring and a regular ring. Lastly, we talk about the zero divisor graph on $C_c(X)_F$., Comment: 19 pages. arXiv admin note: text overlap with arXiv:2109.06414

Let $\mathcal{P}$ be an ideal of closed subsets of a topological space $X$. Consider the ring, $C(X)_\mathcal{P}$ of real valued functions on $X$ whose closure of discontinuity set is a member of $\mathcal{P}$. We investigate the ring properties of $C(X)_\mathcal{P}$ for different choices of $\mathcal{P}$, such as the $\aleph_0$-self injectivity and regularity of the ring, if and when the ring is Artinian and/or Noetherian. The concept of $\mathcal{F}P$-space was introduced by Z. Gharabaghi, M. Ghirati and A. Taherifar in 2018 in a paper published in Houston Journal of Mathematics. In this paper, they established a result stating that every $P$-space is a $\mathcal{F}P$-space. We furnish that this theorem might fail if $X$ is not Tychonoff and we provide a suitable counter example to prove our assertion.

In this paper, we study some properties of the ring $C(X)_F$ of all real valued functions which are continuous except on some finite subsets of $X$. We show that $C(X)_F$ is closed under uniform limit if and only if the set of all non-isolated points of $X$ is finite. We also initiate and investigate the zero divisor graph of the ring $C(X)_F$.

A space $X$ is called $CCS$-normal space if there exist a normal space $Y$ and a bijection $f: X\mapsto Y$ such that $f\lvert_C:C\mapsto f(C)$ is homeomorphism for any cellular-compact subset $C$ of $X$. We discuss about the relations between $C$-normal, $CC$-normal, $Ps$-normal spaces with $CCS$-normal.

We have defined almost separable space. We show that like separability, almost separability is $c$ productive and converse also true under some restrictions. We establish a Baire Category theorem like result in Hausdorff, Pseudocompacts spaces. We investigate few relationships among separability, almost separability, sequential separability, strongly sequential separability.

Let $\Sigma (X,\mathbb{C})$ denote the collection of all the rings between $C^*(X,\mathbb{C})$ and $C(X,\mathbb{C})$. We show that there is a natural correlation between the absolutely convex ideals/ prime ideals/maximal ideals/$z$-ideals/$z^\circ$-ideals in the rings $P(X,\mathbb{C})$ in $\Sigma(X,\mathbb{C})$ and in their real-valued counterparts $P(X,\mathbb{C})\cap C(X)$. It is shown that the structure space of any such $P(X,\mathbb{C})$ is $\beta X$. We show that for any maximal ideal $M$ in $C(X,\mathbb{C}), C(X,\mathbb{C})/M$ is an algebraically closed field. We give a necessary and sufficient condition for the ideal $C_{\mathcal{P}}(X,\mathbb{C})$ of $C(X,\mathbb{C})$ to be a prime ideal, and we examine a few special cases thereafter.

A space $X$ is called $Ps$-normal($Ps$-Tychonoff) space if there exists a normal(Tychonoff) space $Y$ and a bijection $f: X\mapsto Y$ such that $f\lvert_K:K\mapsto f(K)$ is homeomorphism for any pseudocompact subset $K$ of $X$. We establish a few relations between $C$-normal, $CC$-normal, $L$-normal, $C$-Tychonoff, $CC$-Tychonoff spaces with $Ps$-normal and $Ps$-Tychonoff spaces.

Two separated realcompact measurable spaces $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ are shown to be isomorphic if and only if the rings $\mathcal{M}(X,\mathcal{A})$ and $\mathcal{M}(Y,\mathcal{B})$ of all real valued measurable functions over these two spaces are isomorphic. It is furthermore shown that any such ring $\mathcal{M}(X,\mathcal{A})$, even without the realcompactness hypothesis on $X$, can be embedded monomorphically in a ring of the form $C(K)$, where $K$ is a zero dimensional Hausdorff topological space. It is also shown that given a measure $\mu$ on $(X,\mathcal{A})$, the $m_\mu$-topology on $\mathcal{M}(X,\mathcal{A})$ is 1st countable if and only if it is connected and this happens when and only when $\mathcal{M}(X,\mathcal{A})$ becomes identical to the subring $L^\infty(\mu)$ of all $\mu$-essentially bounded measurable functions on $(X,\mathcal{A})$. Additionally, we investigate the ideal structures in subrings of $\mathcal{M}(X,\mathcal{A})$ that consist of functions vanishing at all but finitely many points and functions 'vanishing at infinity' respectively. In particular, we show that the former subring equals the intersection of all free ideals in $\mathcal{M}(X,\mathcal{A})$ when $(X,\mathcal{A})$ is separated and $\mathcal{A}$ is infinite. Assuming $(X,\mathcal{A})$ is locally finite, we also determine a pair of necessary and sufficient conditions for the later subring to be an ideal of $\mathcal{M}(X,\mathcal{A})$., Comment: 13 Pages

The set of all maximal ideals of the ring $\mathcal{M}(X,\mathcal{A})$ of real valued measurable functions on a measurable space $(X,\mathcal{A})$ equipped with the hull-kernel topology is shown to be homeomorphic to the set $\hat{X}$ of all ultrafilters of measurable sets on $X$ with the Stone-topology. This yields a complete description of the maximal ideals of $\mathcal{M}(X,\mathcal{A})$ in terms of the points of $\hat{X}$. It is further shown that the structure spaces of all the intermediate subrings of $\mathcal{M}(X,\mathcal{A})$ containing the bounded measurable functions are one and the same and are compact Hausdorff zero-dimensional spaces. It is observed that when $X$ is a $P$-space, then $C(X) = \mathcal{M}(X,\mathcal{A})$ where $\mathcal{A}$ is the $\sigma$-algebra consisting of the zero-sets of $X$., Comment: 15 pages, 0 figures

For any ideal $\mathcal{P}$ of closed sets in $X$, let $C_\mathcal{P}(X)$ be the family of those functions in $C(X)$ whose support lie on $\mathcal{P}$. Further let $C^\mathcal{P}_\infty(X)$ contain precisely those functions $f$ in $C(X)$ for which for each $\epsilon >0, \{x\in X: \lvert f(x)\rvert\geq \epsilon\}$ is a member of $\mathcal{P}$. Let $\upsilon_{C_{\mathcal{P}}}X$ stand for the set of all those points $p$ in $\beta X$ at which the stone extension $f^*$ for each $f$ in $C_\mathcal{P}(X)$ is real valued. We show that each realcompact space lying between $X$ and $\beta X$ is of the form $\upsilon_{C_\mathcal{P}}X$ if and only if $X$ is pseudocompact. We find out conditions under which an arbitrary product of spaces of the form locally-$\mathcal{P}/$ almost locally-$\mathcal{P}$, becomes a space of the same form. We further show that $C_\mathcal{P}(X)$ is a free ideal ( essential ideal ) of $C(X)$ if and only if $C^\mathcal{P}_\infty(X)$ is a free ideal ( respectively essential ideal ) of $C^*(X)+C^\mathcal{P}_\infty(X)$ when and only when $X$ is locally-$\mathcal{P}$ ( almost locally-$\mathcal{P}$). We address the problem, when does $C_\mathcal{P}(X)/C^\mathcal{P}_{\infty}(X)$ become identical to the socle of the ring $C(X)$. Finally we observe that the ideals of the form $C_\mathcal{P}(X)$ of $C(X)$ are no other than the $z^\circ$-ideals of $C(X)$.

A proper ideal $I$ in a commutative ring with unity is called a $z^\circ$-ideal if for each $a$ in $I$, the intersection of all minimal prime ideals in $R$ which contain $a$ is contained in $I$. For any totally ordered field $F$ and a completely $F$-regular topological space $X$, let $C(X,F)$ be the ring of all $F$-valued continuous functions on $X$ and $B(X,F)$ the aggregate of all those functions which are bounded over $X$. An explicit formula for all the $z^\circ$-ideals in $A(X,F)$ in terms of ideals of closed sets in $X$ is given. It turns out that an intermediate ring $A(X,F)\neq C(X,F)$ is never regular in the sense of Von-Neumann. This property further characterizes $C(X,F)$ amongst the intermediate rings within the class of $P_F$-spaces $X$. It is also realized that $X$ is an almost $P_F$-space if and only if each maximal ideal in $C(X,F)$ is $z^\circ$-ideal. Incidentally this property also characterizes $C(X,F)$ amongst the intermediate rings within the family of almost $P_F$-spaces.