Twisted Polynomial and Power Series Rings

Abstract

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.

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Correspondence to Phan Thanh Toan.

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This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2019.06.

Communicated by Fariborz Azarpanah.

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Chang, G.W., Toan, P.T. Twisted Polynomial and Power Series Rings. Bull. Iran. Math. Soc. (2021). https://doi.org/10.1007/s41980-020-00503-5

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Keywords

  • Twisted polynomial
  • Twisted power series
  • UFD
  • GCD-domain

Mathematics Subject Classification

  • 13A05
  • 13A15
  • 13B25
  • 13F25
  • 13G05