Some Ring and Module Properties of Skew Power Series

Abstract

All rings are assumed to be associative and with nonzero identity element. Let ϕ be an injective endomorphism of a ring A. We denote by A _ℓ [[x, ϕ]] the left skew power series ring consisting of formal series $\sum\nolimits_{i = 0}^\infty {{a_i}{x^i}}$ of an indeterminate x with canonical coefficients a _i ∈ A, where addition is defined naturally and multiplication is defined by the rule x ⁱ a = ϕ ⁱ(a)x ⁱ The ring A _ℓ [[x, ϕ]] contains the left skew polynomial ring A _ℓ [x, ϕ]. For every right A-module M, we denote by M _ℓ [[x, ϕ]] the set of all formal sums of the form $f \equiv \sum\nolimits_{i = 0}^\infty {{m_i}{x^i}}$, where m _i ∈ M. The coefficient fo is the constant term of the series f ₀. It is directly verified that the set M _ℓ [[x, ϕ]] is a right module over the left skew power series ring A _ℓ [[x, ϕ]] such that addition is defined naturally and multiplication by elements of the ring A _ℓ [[x, ϕ]] is defined by the rule $(\sum\nolimits_{i = 0}^\infty {{m_i}{x^i})(\sum\nolimits_{j = 0}^\infty {{a_j}{x^j}} )}\mathop {}\limits_{}{ = _{}}\sum\nolimits_{k = 0}^\infty {(\sum\nolimits_{i + j = k}{{m_j}{\varphi ^i}({a_j})){x^k}} }$. The right A _ℓ [[x, ϕ]-module M _ℓ [[x, ϕ]] is called the skew power series module.