Let R be a commutative ring with identity and \({\mathbb {N}}_0\) be the additive monoid of nonnegative integers. We say that a function \(t : {\mathbb {N}}_0 \times {\mathbb {N}}_0 \rightarrow R\) is a twist function on R if t satisfies the following three properties for all \(n, m, q \in {\mathbb {N}}_0\): (i) \(t(0,q) = 1\), (ii) \(t(n,m) = t(m,n)\), and (iii) \(t(n,m) \cdot t(n + m, q) = t (n, m + q) \cdot t(m, q)\). Let \(R[\![X]\!]\) (resp., R[X]) be the set of power series (resp., polynomials) with coefficients in R. For \(f = \sum _{n=0}^{\infty } a_nX^n\) and \(g = \sum _{n=0}^{\infty } b_nX^n \in R[\![X]\!]\), let \(f+g = \sum _{n=0}^{\infty } (a_n+b_n)X^n\), \(f*_tg = \sum _{n=0}^{\infty }(\sum _{i+j = n}t(i,j)a_ib_j)X^n\). Then, \(R^t[\![X]\!]:= (R[\![X]\!], +, *_t)\) and \(R^t[X] := (R[X], +, *_t)\) are commutative rings with identity that contain R as a subring. In this paper, we study ring-theoretic properties of \(R^t[\![X]\!]\) and \(R^t[X]\) 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.

令R为具有同一性的可交换环，\（{\ mathbb {N}} _ 0 \）为非负整数的加和对映体。我们说一个函数\（T：{\ mathbb {N}} _ 0 \倍{\ mathbb {N}} _ 0 \ RIGHTARROW r \）是扭曲函数ř如果吨满足以下的所有三个属性\（正，m，q \ in {\ mathbb {N}} _ 0 \）中：（i）\（t（0，q）= 1 \），（ii）\（t（n，m）= t（m，n ）\）和（iii）\（t（n，m）\ cdot t（n + m，q）= t（n，m + q）\ cdot t（m，q）\）。令\（R [\！[X] \！] \）（resp。，R [ X ]）是幂级数集（resp。，多项式），其系数为[R 。对于\ [f = \ sum _ {n = 0} ^ {\ infty} a_nX ^ n \）和\（g = \ sum _ {n = 0} ^ {\ infty} b_nX ^ n \ in R [\！ [X] \！] \），令\（f + g = \ sum _ {n = 0} ^ {\ infty}（a_n + b_n）X ^ n \），\（f * _tg = \ sum _ { n = 0} ^ {\ infty}（\ sum _ {i + j = n} t（i，j）a_ib_j）X ^ n \）。然后，\（R ^ t [\！[X] \！]：==（R [\！[X] \！]，+，* _t）\）和\（R ^ t [X]：=（R [X]，+，* _ t）\）是具有R子环身份的可交换环。在本文中，我们研究\（R ^ t [\！[X] \！] \）和\（R ^ t [X] \）的环理论性质着重于除数属性，包括UFD和GCD域。我们还展示了这两个环与通常的幂级数环和多项式环之间的关系。