$\omega$-Euclidean domain and Laurent series

Abstract

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

$\omega$-Euclidean domain and Laurent series

“…Also K. Amano and T. Kuzumaki in [1] proved that if R is a right Euclidean domain then a skew Laurent formal series ring is a Euclidean domain. O. M. Romaniv and A. V. Sagan in [4] proved that R is a commutative ω-Euclidean domain if and only if a Laurent formal series ring is a ω-Euclidean domain. In this paper, the results in [5] and [4] for a skew Laurent formal series ring are generalized.…”

“…O. M. Romaniv and A. V. Sagan in [4] proved that R is a commutative ω-Euclidean domain if and only if a Laurent formal series ring is a ω-Euclidean domain. In this paper, the results in [5] and [4] for a skew Laurent formal series ring are generalized. In addition, elementary reduction of matrices over a skew Laurent formal series ring are investigated.…”