## Sums of exceptional units in finite commutative rings

### Volume 188 / 2019

Acta Arithmetica 188 (2019), 317-324
MSC: Primary 11T30; Secondary 11T99.
DOI: 10.4064/aa170131-23-8
Published online: 18 March 2019

#### Abstract

For a finite commutative ring $R$ with identity, we obtain an exact formula for the number of ways to represent each element of $R$ as the sum of two exceptional units. This generalizes to finite rings a recent result of J. W. Sander for the ring $\mathbb Z_n$ of residue classes mod $n$. We also obtain a formula for the number of exceptional units in $R$, generalizing to finite rings a result of Harrington and Jones in $\mathbb Z_n$.