ON A STRONGLY Μ-NC RING

Abstract

This work aims to investigate and advance the concept of strongly M-nil clean (S-Μ-NC) rings. In these rings, each element can be expressed as the sum of a nilpotent element and a Μ-potent element that commute. The study focuses on:

Defining and formalizing the notions of highly Μ-NC rings and S-Μ-NC elements.

Proving that if 2 is nilpotent in a strongly S-Μ-NC ring, then any element

????∈????  with the formula ????2-a is also nilpotent.

Investigating the role of M-units in such rings, particularly when the ring’s characteristic

char(????)

 demonstrating that, under these conditions, each ideal of ???? is a Μ-potent lift module ideal.

This study enhances the understanding of clean rings with specific algebraic and commutativity constraints, providing new insights into the interaction between nilpotent and Μ-potent elements and their potential applications in algebraic research.