On the sumsets of exceptional units in quaternion rings

2024/06/05 by Hassan Cheraghpour, Cheraghpour, Hassan, David Dolžan +1 · 3 citations
Mathematics · #Graph theory and applications #Algebraic and Geometric Analysis #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2406.03312

Abstract

We investigate sums of exceptional units in a quaternion ring H(R) over a finite commutative ring R. We prove that in order to find the number of representations of an element in H(R) as a sum of k exceptional units for some integer k ≥ 2, we can limit ourselves to studying the quaternion rings over local rings. For a local ring R of even order, we find the number of representations of an element of H(R) as a sum of k exceptional units for any integer k ≥ 2. For a local ring R of odd order, we find either the number or the bounds for the number of representations of an element of H(R) as a sum of 2 exceptional units.

Cited by

• Unipotent orbits of elements in a quaternion ring of odd order
• Products of idempotents in a quaternion ring
• Products of nilpotents in a quaternion ring of odd order

Related

• On the sumsets of exceptional units in ℤ_n
• On the additive and multiplicative structures of the exceptional units in finite commutative rings
• Products of nilpotents in a quaternion ring of odd order
• On the sums of squares of exceptional units in residue class rings
• A Note on Quaternions over Commutative Rings