Một R vòng được gọi là yếu tổng quát abelian (đối với ngắn hạn, W GA-ring) nếu cho mỗi e idempotent trong R, có tồn tại idempotents f, g, h trong R như ER ~ e R ⊕ gr = (1 - e)R ~ e R ⊕ ân sự, trong khi gr và nhân sự không có khác không summands đẳng cấu. = Bằng một ví dụ, chúng tôi sẽ cho thấy. | Vietnam Journal of Mathematics 34 2 2006 171-178 Viet n a m J o u r n a I of MATHEMATICS VAST 2006 K0 of Exchange Rings with Stable Range 1 Xinmin Lu1 2 and Hourong Qin2 1 Faculty of Science Jiangxi University of Science and Technology Ganzhou 341000 P. R. China 2Department of Mathematics Nanjing University Nanjing 210093 China Received January 28 2005 Revised February 28 2006 Abstract. A ring is called weakly generalized abelian for short -ring if for each idempotent in there exist such that . and. while and have no isomorphic nonzero summands. By an example we will show that the class of generalized abelian rings for short -rings introduced in 10 is a proper subclass of the class of -rings. We will prove that for an exchange ring with stable range 1 0 is an -group if and only if is a -ring. 2000 Mathematics subject classification 19A49 16E20 06F15. Keywords 0-group exchange ring weakly generalized Abelian ring Stable range 1 -group. 1. Introduction If is a unit-regular ring is 0 R torsion-free and unperforated The research was partially supported by the NSFC Grant and the second author was partially supported by the National Distinguished Youth Science Foundation of China Grant and the 973 Grant. 172 Xinmin Lu and Hourong Qin 0 0. .0. . . .0. .0. . K 0 R . . .-ring. .GAERS-1. . . GAERS-1 0. Under what condition o of an exchange ring with stable range 1 is torsion-free 0 2. Preliminaries 0 0 Rings and modules. . .directly finite . n. . m m. .stable range 1. . . .exchange . . A i. R. 0 of Exchange Rings with Stable Range 1 173 . - A . iei . isomorphic BB A B BB R B B BB R BBB BBBB BBBBBBBB BBBBBBB B . W . . 0 B . BB B BBBB 0 B BB B B BB B BBBBB BBBB B 0 B .0 B B . BB B BBBBBB B BB B BkB 0 B BBBB B BBB BB BB . BBBBB BBB BB BB BBBB .0 B .0. B 0 B .0. BB B BBBB B 0 B BB B B BB BB B 0 B BBBBBBBB BBB . pre-order BB 0 B -groups. .least upper bound BB .greatest lower bound. .upper semilattice. .
