The Source Of Primeness Of Rings

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.