Unit 1-stable range for ideals

Let R be a ring with identity 1. We say that R satisfies unit 1-stable range in case ax+b = 1 with a,x,b ∈ R implying that a+bu ∈ U(R). Many authors studied unit 1-stable range such as those of [1, 2, 3, 4, 5, 6]. Following the authors, a ring R satisfies unit 1-stable range for an ideal I provided that ax+b = 1 with a∈ I, x,b ∈ R implying that x+ub ∈U(R) for some unit u∈U(R). Let R = Z/2Z⊕Z/3Z and I = 0⊕Z/3Z. Then R satisfies unit 1-stable range for I, while in fact R does not satisfy unit 1-stable range. Thus the concept of unit 1-stable range for an ideal is a nontrivial generalization of that of rings satisfying such stable range condition. In this note, we investigate necessary and sufficient conditions under which a ring R satisfies unit 1-stable range for an ideal. It is shown that R satisfies unit 1-stable range for I if and only if QM2(R) satisfies unit 1-stable range for QM2(I). Throughout, all rings are associative with identity. Mn(R) denotes the ring of n×n matrices over R, GLn(R) denotes the n-dimensional general linear group of R, TMn(R) denotes the ring of all n×n lower triangular matrices over R, and TMn(I) denotes the ideal of all n×n lower triangular matrices over I. Clearly, TMn(I) is an ideal of TMn(R). We begin with the following.