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A NOTE ON RINGS OF WEAKLY STABLE RANGE ONE

Huanyin Chen, Miaosen Chen

2006
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NEW ZEALAND JOURNAL OF MATHEMATICS
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unpublished

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
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... ular 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 use M n (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 End R (A) = 1. Then wsr End R (nA) = 1 for all n ∈ N. Proof. Given M = A 1 ⊕ B = A 2 ⊕ C with A 1 ∼ = nA ∼ = A 2 , we have M = A 11 ⊕ · · · ⊕ A 1n ⊕ B = A 21 ⊕ · · · ⊕ A 2n ⊕ C with A 1i ∼ = A ∼ = A 2i for all i. As wsr End R (A) = 1, by [3, Proposition 2], we can find some D 1 , E 1 ⊆ M such that M = D 1 ⊕ E 1 ⊕ (A 12 ⊕ · · · ⊕ A 1n ⊕ B) = D 1 ⊕ (A 22 ⊕ · · · ⊕ A 2n ⊕ C) or M = D 1 ⊕ (A 12 ⊕ · · · ⊕ A 1n ⊕ B) = D 1 ⊕ E 1 ⊕ (A 22 ⊕ · · · ⊕ A 2n ⊕ C). Thus we get M = (E 1 ⊕ A 12) ⊕ (A 13 ⊕ · · · ⊕ A 1n ⊕ B ⊕ D 1) = A 22 ⊕ (A 23 ⊕ · · · ⊕ A 2n ⊕ C ⊕ D 1) or M = A 12 ⊕(A 13 ⊕· · ·⊕A 1n ⊕B ⊕D 1) = (E 1 ⊕A 22)⊕(A 23 ⊕· · ·⊕A 2n ⊕C ⊕D 1). As a result, we get M = A 12 ⊕(A 13 ⊕· · ·⊕A 1n ⊕B⊕D 1) = A 22 ⊕(A 23 ⊕· · ·⊕A 2n ⊕C⊕D 1), where A 12 = E 1 ⊕ A 12 or A 12 = A 12 and A 22 = A 22 or A 22 = E 1 ⊕ A 22. Clearly, 1991 Mathematics Subject Classification 16U99.

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