群环的幂等稳定度1

Group rings of idempotent stable range one

设a∈R,如果对环R元素b,满足aR+bR=R,则存在幂等元e∈R,使得a+be有左逆,那么称元素a有幂等稳定度1(记为isr(a)=1).如果对于R中的所有元素a,都有isr(a)=1,那么称环R有幂等稳定度1(记为isr(R)=1).证明了若R是半完全环,G是初等阿贝尔p-群,则isr(RG)=1.另外,若isr(R)=1,G是局部有限p-群,且p∈J(G),则isr(RG)=1.

An element a in a ringR is said to have idempotent stable range 1 （written isr（ a ） = 1） if aR ＋bR = R （for any b ∈ R ） implies there exists an idempotent e ∈ R , such that a＋beis left invertible. If isr（ a ） = 1 , for all a E R , then R, is called to have idempotent stable range 1 （written isr（ R ） = 1 ）. In this paper, we showed that i{ R was semiperfect and G was an elementary abelian p -group, then isr（ RG ） = 1. It was shown that if isr（ R ） = 1 and p ∈ J （G） and G was a locally finite p -group, then isr（ RG ） = 1.