Element absorb Topology on rings

TL;DR

This paper introduces a new topology on rings based on element absorption, linking algebraic properties with topological structures, and explores its implications for Pierce decomposition.

Contribution

It defines a novel topology related to elements in rings and connects it to algebraic decompositions like Pierce decomposition.

Findings

Topology characterizes algebraic properties of rings.

Provides a topological perspective on Pierce decomposition.

Links idempotent elements to topological structures.

Abstract

In this paper, we introduce a new Topology related to special elements in a noncummutative rings. Consider a ring $R$, we denote by $\textrm{Id}(R)$ 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 $\tau_a:=\{ I\subseteq R | Ia \subseteq I\} \subseteq P(R)$. 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, $\tau_e:=\{ I\subseteq R | Ie \subseteq I\} \subseteq P(R)$. We present Topological description of the pierce decomposition $ R=Re\oplus R(1-e)$.