$\omega$-Euclidean domain and skew Laurent series rings

TL;DR

This paper investigates the properties of skew Laurent formal series rings over right $$-Euclidean domains, establishing conditions under which these rings inherit Euclidean and principal ideal domain properties, and analyzing matrix reduction features.

Contribution

It proves that skew Laurent formal series rings over right $$-Euclidean domains are also right $$-Euclidean, and under certain conditions, they are principal ideal domains and have elementary matrix reduction.

Findings

Skew Laurent formal series rings over right $$-Euclidean domains are right $$-Euclidean.

If the base domain has a multiplicative norm, the skew Laurent series ring is a principal ideal domain.

Noncommutative $$-Euclidean domains with a multiplicative norm have rings with elementary matrix reduction.

Abstract

In this paper we proved that if $R$ is right $\omega$-Euclidean domain, then skew Laurent formal series ring is right $\omega$-Euclidean domain. We also showed that if $R$ is a right $\omega$-Euclidean domain with multiplicative norm, then skew Laurent formal series ring is a right principal ideal domain. In addition, we proved that if $R$ is a noncommutative $\omega$-Euclidean domain with a multiplicative norm, then $R$ and skew Laurent formal series ring is a ring with elementary reduction of matrices.