Abstract
In this paper, we show that a ring R satisfies unit 1-stable range if and only if a1R + ⋯ + amR = dR with m ≥ 2,a 1, ⋯am ϵR implies that there exist u1 , ⋯um ϵ U(R) such that a1u1 +⋯+amum = d and an exchange ring R has stable range one if and only if a1R+⋯+amR = dR with m ≥ 2,a 1,⋯,am ϵ R implies that there exist unit-regular w 1,⋯,wm ϵ R such that a1w1 +⋯+ amwm = d. Also we show that an exchange ring R satisfies the n-stable range condition if and only if a( nR)+bR = dR with a ϵ Rn,b,d ϵ R implies that there exist unimodular regular w ϵ n R and: y ϵ R such that aw+by = d.