Some Ring and Module Properties of Skew Laurent Series

Diar A. Tuganbaev
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Diar A. Tuganbaev: Moscow State University

A chapter in Formal Power Series and Algebraic Combinatorics, 2000, pp 613-622 from Springer

Abstract: 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

Keywords: Lower Coefficient; Division Ring; Laurent Series; Series Ring; Artinian Ring (search for similar items in EconPapers)
Date: 2000
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DOI: 10.1007/978-3-662-04166-6_60

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