Some results on \(n\)-stable rings (Q1407483)

Let \(R\) be a commutative ring with identity. For any fixed integers \(n\geq 1\) and \(1\leq k\leq n\), a sequence \(a_1,\dots,a_{s+1}\) of elements of \(R\) with \(s\geq n\) is said to be \((n,k)\)-stable if \((a_1,a_2,\dots, a_{s+1})=(a_1+b_1a_{s+1}, a_2+ b_2a_{s+1},\dots,a_k+b_k a_{s+1},a_{k+1},\dots,a_s)\) for some \(b_1,\dots,b_k\in R\). \(R\) is an \((n,k)\)-stable ring if any unimodular sequence of size larger than \(n\) is \((n,k)\)-stable. In this paper, the author studies \((n,k)\)-stable rings.