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

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.