Exchange Rings Having Stable Range One

We investigate the sufficient conditions and the necessary conditions on an exchange ring R under which R has stable range one. These give nontrivial generalizations of Theorem 3 of V. P. Camillo and H.-P. Yu (1995), Theorem 4.19 of K. R. Goodearl (1979, 1991), Theorem 2 of R. E. Hartwig (1982), and Theorem 9 of H.-P. Yu (1995). 2000 Mathematics Subject Classification. Primary 16E50, 19B10. An associative ring R is called exchange if for every right R-module A and any two decompositions A=M⊕N =i∈I Ai, where MR R and the index set I is finite, there exist submodules Ai ⊆ Ai such that A = M ⊕ ( ⊕ i∈I Ai). We know that regular rings, π -regular rings, strongly π -regular rings, semiperfect rings, left or right continuous rings, clean rings, unit C∗-algebras of real rank zero [2, Theorem 4.2], right semiartinian rings, and the ring of all ω×ω matrices over regular R, which are both row and colume-finite, are exchange rings. We callR has stable range one provided thataR+bR = R implies thata+by ∈U(R) for a y ∈ R. It is well known that an exchange ring R has stable range one if and only if A⊕B A⊕C implies B C for all finitely generated projective right R-modules A, B, and C . It has been realized that the class of rings having stable range one has good stability properties in a K-theoretic sense (cf. [8, 13]). Many authors have studied stable range one conditions over exchange rings such as [1, 2, 4, 5, 6, 7, 15, 16]. In this paper, we investigate stable range one conditions over exchange rings by virtue of Drazin inverses, nilpotent elements, and prime ideals. We showed that stable range conditions can be determined by Drazin inverses for exchange rings. Also, we see that these stable range conditions can be determined by regular elements out of any proper ideal of R. Moreover, we prove that an exchange ring R has stable range one if the set of nilpotents is closed under product. These extend the corresponding results of [4, Theorem 3], [9, Theorem 4.19], [12, Theorem 2], and [16, Theorem 9]. Throughout this paper, all rings are associative ring with identities and all right R-modules are unitary right R-modules. M ⊕ N means that right R-module M is isomorphic to a direct summand of right R-module N . The notation x ≈y means that x = uyu−1 for some u ∈ U(R), where U(R) denotes the set of all units of R. Call a ∈ R is regular if a = axa for some x ∈ R and a ∈ R is unit-regular if a = aua for some u∈U(R). 1. Drazin inverses. Recall that a ∈ R is called strongly π -regular if there exist n ≥ 1 and x ∈ R such that an = an+1x, ax = xa, and x = xax. By [1, Theorem 3], we know that every strongly π -regular element of an exchange ring is unit-regular.