On the sumsets of exceptional units in quaternion rings
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 $k \geq 2$, 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 $k \geq 2$. 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2024
 DOI:
 10.48550/arXiv.2406.03312
 arXiv:
 arXiv:2406.03312
 Bibcode:
 2024arXiv240603312C
 Keywords:

 Mathematics  Rings and Algebras