Conjugate (nil) clean rings and Kothe's problem

TL;DR

This paper explores the relationship between nil clean rings, conjugate clean rings, and Kothe's problem, establishing equivalences and introducing new classes of rings to deepen understanding of ring decomposition properties.

Contribution

It introduces conjugate (nil) clean rings, investigates their properties, and links the nil clean matrix ring problem to Kothe's problem over F_2, providing new insights into ring theory.

Findings

Positive answer to matrix nil clean question is equivalent to Kothe's problem over F_2.

Introduces and studies conjugate clean and conjugate nil clean rings.

Establishes new classes of rings between uniquely clean and clean rings.

Abstract

Question 3 of [3] asks whether the matrix ring Mn(R) is nil clean, for any nil clean ring R. It is shown that positive answer to this question is equivalent to positive solution for Kothe's problem in the class of algebras over the field F_2. Other equivalent problems are also discussed. The classes of conjugate clean and conjugate nil clean rings, which lie strictly between uniquely (nil) clean and (nil) clean rings are introduced and investigated.