Uniquely clean 2 × 2 invertible integral matrices

Abstract

While units in any unital ring are strongly clean by denition, which
units are uniquely clean, is a far from being simple question, even in particular
rings. In this paper, the question is solved for 2 2 integral matrices. It turns
out that uniquely clean invertible matrices are scarce: only the matrices similar
to [1 0; 0 -1]. The study is splitted into three cases: the elliptic, the parabolic
and the hyperbolic cases, according to the discriminant of their characteristic
polynomial. In the rst two cases, units are not uniquely clean.