The Goodearl-Menal condition on rings \(R\) states that the element \(a\in R\) satisfies this condition if, for every \(x\in R\), there exists an invertible element \(u\in R\) such that the elements \(x-u\) and \(a-u^{-1}\) both invert in \(R\).
The author of the paper under review describes the \(2\times 2\) and \(3\times 3\) matrices with unit stable range one over arbitrary commutative rings (see Theorems 8, 11 and 13, respectively). Moreover, in Theorems 15 and 17, he characterizes the \(2\times 2\) matrices which satisfy the mentioned above Goodearl-Menal condition. Likewise, for \(2\times 2\) matrices over the ring of integers \(\mathbb{Z}\), the author shows that the stable range one and the unit stable range one properties are completely equivalent, and also that the only matrix which satisfies the Goodearl-Menal condition is the zero matrix.
The paper ends with several left-open problems that are closely related to the explored subject.