{"abstract":"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, \u03c8, \u03c6) 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 \u2208 R such that a + by \u2208 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, \u03c8, \u03c6). In addition, we prove that if T is the ring of a Morita context (R, S, M, N, \u03c8, \u03c6) 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 \u2208 R such that a + by \u2208 R is invertible. Throughout, all rings are associative with identity. We use M n (R) to denote the ring of all n \u00d7 n matrices over the ring R. We use N to denote the set of all natural numbers. The notation A \u2295 B means that A is isomorphic to a direct summand of B. We write R \u2248 S to denote that the rings R and S are Morita equivalent. For any n \u2265 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 \u2208 N. Proof. Given M = A 1 \u2295 B = A 2 \u2295 C with A 1 \u223c = nA \u223c = A 2 , we have M = A 11 \u2295 \u00b7 \u00b7 \u00b7 \u2295 A 1n \u2295 B = A 21 \u2295 \u00b7 \u00b7 \u00b7 \u2295 A 2n \u2295 C with A 1i \u223c = A \u223c = A 2i for all i. As wsr End R (A) = 1, by [3, Proposition 2], we can find some D 1 , E 1 \u2286 M such that M = D 1 \u2295 E 1 \u2295 (A 12 \u2295 \u00b7 \u00b7 \u00b7 \u2295 A 1n \u2295 B) = D 1 \u2295 (A 22 \u2295 \u00b7 \u00b7 \u00b7 \u2295 A 2n \u2295 C) or M = D 1 \u2295 (A 12 \u2295 \u00b7 \u00b7 \u00b7 \u2295 A 1n \u2295 B) = D 1 \u2295 E 1 \u2295 (A 22 \u2295 \u00b7 \u00b7 \u00b7 \u2295 A 2n \u2295 C). Thus we get M = (E 1 \u2295 A 12) \u2295 (A 13 \u2295 \u00b7 \u00b7 \u00b7 \u2295 A 1n \u2295 B \u2295 D 1) = A 22 \u2295 (A 23 \u2295 \u00b7 \u00b7 \u00b7 \u2295 A 2n \u2295 C \u2295 D 1) or M = A 12 \u2295(A 13 \u2295\u00b7 \u00b7 \u00b7\u2295A 1n \u2295B \u2295D 1) = (E 1 \u2295A 22)\u2295(A 23 \u2295\u00b7 \u00b7 \u00b7\u2295A 2n \u2295C \u2295D 1). As a result, we get M = A 12 \u2295(A 13 \u2295\u00b7 \u00b7 \u00b7\u2295A 1n \u2295B\u2295D 1) = A 22 \u2295(A 23 \u2295\u00b7 \u00b7 \u00b7\u2295A 2n \u2295C\u2295D 1), where A 12 = E 1 \u2295 A 12 or A 12 = A 12 and A 22 = A 22 or A 22 = E 1 \u2295 A 22. Clearly, 1991 Mathematics Subject Classification 16U99.","author":[{"family":"Chen"},{"family":"Chen"}],"id":"unknown","issued":{"date-parts":[[2006]]},"title":"A NOTE ON RINGS OF WEAKLY STABLE RANGE ONE","type":"article-journal","volume":"35"}