Some Ring and Module Properties of Skew Laurent Series

Abstract

All rings are assumed to be associative and with nonzero identity element. Let ϕ be an injective automorphism of a ring A. We denote by A((x, ϕ)) the (left) skew Laurent series ring consisting of formal series \(f = \sum\nolimits_{i = k}^\infty {{f_i}{\chi ^i}} \) of an indeterminate x with an integer k (which could be negative) and canonical coefficients f i ∈ A. Addition in M((x, ϕ)) is defined naturally and multiplication is defined by the rule x i a = ϕ i (a)x (∀a ∈ A). The ring A((x)) ≡ A((x, 1 A )) is the ordinary Laurent series ring. For every right A-module M, we denote by A((x, ϕ)) the right A((x, ϕ))-module consisting of skew Laurent series with coefficients in M A . The module M((x, ϕ)) consists of formal series \(\sum\nolimits_{i = k}^\infty {{m_i}{\chi ^i}} \) with coefficients m i ∈ M A . Addition in M((x, ϕ)) is defined naturally and multiplication is defined by the rule (mx)a = (mϕ(a))x (∀m ∈ M, ∀a ∈ A). In particular, the ring A((x, ϕ)) is the skew Laurent series right module with coefficients in A A . For a series f and two its coefficients f i , f j , we say that f i is lower than f j if i < j.