On trinomial units of the group ring RCn
Year: 2025
Type: article
Abstract: Let R be an associative ring with identity, Cn be the cyclic group of order n, and g be a generator of Cn. In this manuscript, for i ∈ N such that 1 ≤ i < n and gcd(i, n) = 1, the algebraic properties of the trinomial elements hi = -1 + g + g-i in the group ring RCn are investigated. More precisely, necessary and sufficient conditions are given for an element hi to be a unit in RCn. When hi is a unit, explicit formulas for computing its inverse are provided. In addition, appropriate upper bounds for the orders of the elements hi in group rings Rq Cp are given when p is a prime number and Rq is a ring of prime characteristic q. Those results are extended to rings RCp where p is a prime number and R satisfies some mild conditions. When p = 5 and p = 7, recurrence formulas for the coefficients of the powers of hi in the ring RCp are established. Moreover, upper bounds for orders of symmetric units with augmentation 1 in group rings Rq Cp are established when Rq satisfies some additional conditions.
Source: TURKISH JOURNAL OF MATHEMATICS
Authors: CÉSAR A. HERNANDEZ MELO, FERNANDA DINIZ de MELO HERNANDEZ, PATRICIA MASSAE KITANI
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W7114928949
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