The Source Of Primeness Of Rings

TL;DR

This paper introduces the concept of the source of primeness in rings, defined via a specific subset, and explores its properties along with those of related elements like idempotents and nilpotents.

Contribution

It defines the source of primeness for rings and investigates its fundamental properties and relationships with key ring elements.

Findings

Properties of the source of primeness in rings analyzed

Relationships between the source of primeness and idempotent, nilpotent, zero divisor elements studied

Basic properties of the source of primeness established

Abstract

Let $R$ be an associative ring. We define a subset $S_{R}^{a}$, where $a\in R$ of $R$ as $S_{R}^{a}=\{b\in R \mid aRb=(0)\}$. Then, the set $P_{R} = \bigcap_{a\in R} S_{R}^{a}$ call it the source of primeness of $R$. We first examine some basic properties of the subset $P_{R}$ in any ring $R$, and properties of idempotent elements, nilpotent elements, zero divisor elements and identity elements. And we investigated the properties of the elements of the set source of primeness with the help of these elements.