Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algebraic properties of the ring $C(X)_\mathcal{P}$

Amrita Dey, Sagarmoy Bag, Dhananjoy Mandal

Abstract

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.

Algebraic properties of the ring $C(X)_\mathcal{P}$

Abstract

Our aim is to study certain algebraic properties of the ring of real-valued functions on whose closure of discontinuity set is in an ideal of closed sets. We characterize -spaces using -ideals and essential ideals of and also almost -spaces using -ideals of and a topology finer than the original topology on . We deduce that each maximal ideal of \cite{GGT2018} (resp. \cite{A2010}) is a -ideal. We establish that the notions of clean ring, weakly clean ring, semiclean ring, almost clean ring and exchange ring coincide in the ring . End of this paper, we also characterize -spaces and almost -spaces using certain ideals having depth zero. We exhibit a condition on under which prime and essential ideals of have depth zero.
Paper Structure (7 sections, 39 theorems, 3 equations)

This paper contains 7 sections, 39 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Ideals of $C(X)_\mathcal{P}$.
  3. $z$-ideals of $C(X)_\mathcal{P}$.
  4. $z^0$-ideals of $C(X)_\mathcal{P}$.
  5. Essential ideals of $C(X)_\mathcal{P}$.
  6. When is $C(X)_\mathcal{P}$ clean?
  7. Depth of ideals of $C(X)_\mathcal{P}$

Key Result

Lemma 1.1

DABM Let $\mathcal{P}$ be an ideal of all closed subsets of a $T_1$-space $X$ and $\mathcal{P}'$ be the ideal of all closed subsets of the set of isolated points of $X$. Then $C(X)_\mathcal{P}=C(X)$ if and only if $\mathcal{P}\subseteq \mathcal{P'}$.

Theorems & Definitions (71)

  • Lemma 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Example 2.5
  • ...and 61 more