群环的幂等稳定度1

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.

Group rings of idempotent stable range one

An element a in a ring R 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∈R , then R, is called to have idempotent stable range 1 (written isr(R) = 1).In this paper, we showed that if 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.