Prime Principal Right Ideal Rings

TL;DR

This paper introduces the concept of prime principal right ideal rings (PPRIR), characterizes their properties, and explores their structure, particularly focusing on prime principal right ideal domains (PPRID) within commutative rings.

Contribution

It defines prime principal right ideal rings and prime principal right ideal domains, providing foundational properties and structural insights into these classes of rings.

Findings

Characterization of prime principal right ideals

Properties of prime principal right ideal rings

Conditions for PPRID in commutative rings

Abstract

Let R be a commutative ring with unity $1\in R$. In this article, we introduce the concept of prime principal right ideal rings (\textbf{PPRIR}), A prime ideal P of R is said to be prime principal right ideal (\textbf{PPRI}) is given by $P =\{ ar : r\in R\}$ for some element a. The ring R is said to be prime principal right ideal ring (\textbf{PPRIR}) if every prime ideal of R is a prime principal right ideal (\textbf{PPRI}). A prime principal right ideal ring R is called a prime principal right ideal domain (\textbf{PPRID}) if R is a domain. Several properties and characteristics of prime principal right ideal ring (\textbf{PPRIR}).