Element absorb Topology on rings

Abstract

In this paper, we introduce a new Topology related to special elements in a noncummutative rings. Consider a ring $R$, we denote by $\text{Id}(R)$ the set of all idempotent elements in $R$. Let $a$ is an element of $R$. The element absorb Topology related to $a$ is defined as $\ddot{I}{„}_a:=\{I\subseteq R|Ia\subseteq I\}\subseteq P(R)$. Since this topology is obtained from act of ring, it explains Some of algebraic properties of ring in Topological language .In a special case when $e$ ia an idempotent element, $\ddot{I}{„}_e:=\{I\subseteq R|Ie\subseteq I\}\subseteq P(R)$. We present Topological description of the pierce decomposition $R=Re\oplus R(1-e)$.