On trinomial units of the group ring <em>RC<sub>n</sub></em>

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.