Approximations of Maximal and Principal Ideal

TL;DR

This paper explores the integration of rough set theory with ring theory, specifically focusing on defining and analyzing rough versions of maximal and principal ideals, extending classical concepts in algebra.

Contribution

It introduces the concepts of rough maximal and principal ideals in ring theory, extending traditional ideals using rough set approximations and studying their properties.

Findings

Defined rough maximal and principal ideals in ring theory.

Analyzed properties of upper and lower approximations of these ideals.

Extended classical ideal concepts with rough set theory.

Abstract

In this paper, we will be delving deeper into the connection between the rough theory and the ring theory precisely in the principle and maximal ideal. The rough set theory has shown by Pawlak as good formal tool for modeling and processing incomplete information in information system. The rough theory is based on two concepts the upper approximation of a given set is the union of all the equivalence classes, which are subsets of the set, and the lower approximation is the union of all the equivalence classes, which are intersection with set non-empty. Many researchers develop this theory and use it in many areas. Here, we will apply this theorem in the one of the most important branches of mathematics that is ring theory. We will try to find the rough principal and maximal ideal as an extension of the notion of a principal maximal ideal respectively in ring theory. In addition, we study the properties of the upper and lower approximation of a principal maximal ideal. However, some researchers use the rough theory in the group and ring theory. Our work, is Shaw there are rough maximal and principle ideal as an extension of the maximal and principle ideal respectively. Our result will introduce the rough maximal ideal as an extended notion of a classic maximal ideal and we study some properties of the lower and the upper approximations a maximal ideal.