On UJ-rings

TL;DR

This paper investigates UJ-rings, where all units are of the form 1 + x with x in the Jacobson radical, exploring their properties and how they behave under algebraic constructions, linking to Kothe's problem.

Contribution

It establishes the equivalence between lifting the UJ property to polynomial rings and Kothe's problem for F_2-algebras, providing new insights into ring theory.

Findings

UJ-rings have specific unit structures related to the Jacobson radical

Lifting the UJ property to polynomial rings relates to Kothe's problem

The behavior of UJ-rings under algebraic constructions is characterized

Abstract

UJ-rings are studied, i.e. ring in which all units can be presented in a form 1 + x, for some x\in J(R). The behavior of UJ-rings under various algebraic construction is investigated. In particular, it is shown that the problem of lifting the UJ property from a ring R to the polynomial ring R[x] is equivalent to the Kothe's problem for F_2-algebras.