A variation of stable range one (Q6594882)

The reviewed article is devoted to the examination of a variation of stable range one for an associative unital ring \(R\), concerning its set of all nilpotent elements \(\mathrm{nil}(R)\).\N\NIn fact, an element \(a\) in a ring \(R\) is said to have \textit{left nilpotent stable range 1} if, whenever \(Ra+Rb=R\) for any \(b \in R\), there exists \(r \in \mathrm{nil}(R)\) such that \(a+rb\in U(R)\).\N\NSome interesting results are obtained (see, respectively, Propositions 2.5, 2.6 and 2.7; Propositions 3.2, 3.3 and 3.4; Theorem 4.2, Corollaries 4.5, 4.6) as well as certain valuable examples are provided (see, respectively, Examples 2.4; 3.1; 4.3, 4.4).