Twisted Polynomial and Power Series Rings

Let R be a commutative ring with identity andN0 be the additive monoid of nonnegative integers. We say that a function t : N_0 × N_0 → R is a twist function on R if t satisfies the following three properties for all n,m, q ∈ N0: (i) t(0, q) = 1, (ii) t(n,m) = t(m, n), and (iii) t(n,m) · t(n + m, q) = t(n,m + q) · t(m, q). Let R[[X]] (resp., R[X]) be the set of power series (resp., polynomials) with coefficients in R. For f = ∑∞ ^_n=0 an X^n and g = ∑ ^∞ _n=0 bn X^n ∈ R[[X]], let f + g = ∑ ^∞ _n=0(an + bn)Xn, f ∗t g = ∑^∞ _n=0(∑i+j=n t(i , j )aibj )X^n. Then, Rt [[X]] := (R[[X]],+, ∗t ) and Rt [X] := (R[X],+, ∗t ) are commutative rings with identity that contain R as a subring. In this paper, we study ring-theoretic properties of Rt [[X]] and Rt [X] with focus on divisibility properties including UFDs and GCD-domains.We also show how these two rings are related to the usual power series and polynomial rings.