On stable range conditions

Abstract

In this paper, we show that a ring R satisfies unit 1-stable range if and only if a₁R + ⋯ + a_mR = dR with m ≥ 2,a ₁, ⋯a_m ϵR implies that there exist u₁ , ⋯u_m ϵ U(R) such that a₁u₁ +⋯+a_mu_m = d and an exchange ring R has stable range one if and only if a₁R+⋯+a_mR = dR with m ≥ 2,a ₁,⋯,a_m ϵ R implies that there exist unit-regular w ₁,⋯,w_m ϵ R such that a₁w₁ +⋯+ a_mw_m = 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.