A recursive construction of units in a class of rings

TL;DR

This paper presents a recursive method to lift units from quotient rings to original rings in associative rings with nil ideals, and applies it to various classes of rings including matrix, chain, and group rings.

Contribution

It introduces a recursive construction technique for units in rings with nil ideals, expanding understanding of unit structure in specific ring classes.

Findings

Units of R/N can be lifted to R under mild conditions.

A recursive procedure for identifying lifted units is provided.

Applications include determining units in matrix, chain, and group rings.

Abstract

Let $R$ be an associative ring with identity and let $N$ be a nil ideal of $R$. It is shown that units of $R/N$ can be lifted to units in $R$. Under some mild conditions on the ring, a procedure is given to determine those lifted units in a recursive way. As an application, the units of several classes of rings are determined, including: matrix rings, chain rings, and group rings where the ring is a chain ring. Numerical examples are given illustrating the main results.