A partition of finite rings makes lifting possible

TL;DR

This paper introduces a partitioning method for finite rings based on idempotents, enabling the lifting of various special elements like nilpotents and roots of unity, which advances algebraic structure analysis.

Contribution

It presents a novel partition technique for finite rings that facilitates lifting key elements, a significant step forward in ring theory.

Findings

Partition of finite rings based on idempotents

Method for lifting special elements such as nilpotents and roots of unity

Enhanced understanding of algebraic structures in finite rings

Abstract

We show that every finite ring has a partition, where each block corresponds to one idempotent. Remarkably, this partition provides a way to \emph{lift} a wide variety of special elements such as idempotents, nilpotents, unipotents, roots of unity and regular elements.