A Note on Rings of Weakly Stable Range One

It is shown that if R and S are Morita equivalent rings then R has weakly stable range 1 (written as wsr(R) = 1) if and only if S has. Let T be the ring of a Morita context (R,S,M,N,ψ, φ) with zero pairings. If wsr(R) = wsr(S) = 1, we prove that T is a weakly stable ring. A ring R is said to have weakly stable range one if aR + bR = R implies that there exists a y ∈ R such that a + by ∈ R is right or left invertible. We denote this by wsr(R) = 1. By [2, Proposition 6], it is known that a regular ring R is one–sided unit–regular if and only if wsr(R) = 1. Many authors have studied rings of weakly stable range one, for example [2 –5] and [7–8]. In this note, we investigate equivalent characterizations of weakly stable range one. We prove that if R and S are Morita equivalent rings then wsr(R) = 1 if and only if wsr(S) = 1. This generalizes a corresponding result for one-sided unit–regular rings (cf. [Corollary 7]3). Furthermore, we study weakly stable range one over trivial extensions of rings, power series rings and the ring of a Morita context (R,S,M,N,ψ, φ). In addition, we prove that if T is the ring of a Morita context (R,S,M,N,ψ, φ) with zero pairings and wsr(R) = wsr(S) = 1, then T is a weakly stable ring ( where here sr(R) = 1 indicates that R has stable range one, i.e., aR + bR = R implies that there exists a y ∈ R such that a + by ∈ R is invertible ) . Throughout, all rings are associative with identity. We useMn(R) to denote the ring of all n×n matrices over the ring R. We use N to denote the set of all natural numbers. The notation A .⊕ B means that A is isomorphic to a direct summand of B. We write R ≈ S to denote that the rings R and S are Morita equivalent. For any n ≥ 1 and any module A, we let nA denote the direct sum of n copies of A. Lemma 1. Let A be a right R–module such that wsr ( EndR(A) ) = 1. Then wsr ( EndR(nA) ) = 1 for all n ∈ N. Proof. Given M = A1 ⊕ B = A2 ⊕ C with A1 ∼= nA ∼= A2, we have M = A11 ⊕ · · · ⊕ A1n ⊕ B = A21 ⊕ · · · ⊕ A2n ⊕ C with A1i ∼= A ∼= A2i for all i. As wsr ( EndR(A) ) = 1, by [3, Proposition 2], we can find some D1, E1 ⊆ M such that M = D1 ⊕E1 ⊕ (A12 ⊕ · · · ⊕A1n ⊕B) = D1 ⊕ (A22 ⊕ · · · ⊕A2n ⊕C) or M = D1 ⊕ (A12 ⊕ · · · ⊕A1n ⊕B) = D1 ⊕E1 ⊕ (A22 ⊕ · · · ⊕A2n ⊕C). Thus we get M = (E1⊕A12)⊕ (A13⊕· · ·⊕A1n⊕B⊕D1) = A22⊕ (A23⊕· · ·⊕A2n⊕C⊕D1) or M = A12⊕(A13⊕· · ·⊕A1n⊕B⊕D1) = (E1⊕A22)⊕(A23⊕· · ·⊕A2n⊕C⊕D1). As a result, we getM = A12⊕(A13⊕· · ·⊕A1n⊕B⊕D1) = A22⊕(A23⊕· · ·⊕A2n⊕C⊕D1), where A12 = E1 ⊕ A12 or A12 = A12 and A22 = A22 or A22 = E1 ⊕ A22. Clearly, 1991 Mathematics Subject Classification 16U99.