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- Title
Twisted Polynomial and Power Series Rings.
- Authors
Chang, Gyu Whan; Toan, Phan Thanh
- Abstract
Let R be a commutative ring with identity and N 0 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 ∈ N 0 : (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 ∞ a n X n and g = ∑ n = 0 ∞ b n X n ∈ R [ [ X ] ] , let f + g = ∑ n = 0 ∞ (a n + b n) X n , f ∗ t g = ∑ n = 0 ∞ (∑ i + j = n t (i , j) a i b 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.
- Subjects
POLYNOMIAL rings; POLYNOMIALS; POWER series; COMMUTATIVE rings
- Publication
Bulletin of the Iranian Mathematical Society, 2022, Vol 48, Issue 1, p93
- ISSN
1018-6301
- Publication type
Article
- DOI
10.1007/s41980-020-00503-5