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Turkish Journal of Mathematics

Author ORCID Identifier

CÉSAR A. HERNANDEZ MELO: 0009-0003-7194-4543

FERNANDA DINIZ de MELO HERNANDEZ: 0009-0002-9545-9852

PATRICIA MASSAE KITANI: 0000-0003-1557-6486

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.

DOI

10.55730/1300-0098.3622

Keywords

Group rings, trinomial units, orders

First Page

738

Last Page

754

Publisher

The Scientific and Technological Research Council of Türkiye (TÜBİTAK)

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

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