Abel环的一些刻画(Ⅲ)

Some characterizations on Abel rings(Ⅲ)

设R为环,证明了如下结论:1)R为Abel环当且仅当对任意x,y∈R,当1-xy∈GPE(R)时必有1-yx∈GPE(R);2)若R为正则环,则PE(R)为正则环;3)R为约化环当且仅当对每个e∈E(R),a∈N(R),存在x∈R,使得ae=eaxae;4)R为强正则环当且仅当对任意a,b∈R,存在x∈R,使得ab=baxab.

In this paper, the following results are proved： 1） A ring R is an Abel ring if and only if for each x, y∈R, 1-xy∈GPE（R） implies that 1-yx∈GPE（R）; 2） If R is a regular ring, then PE（R） is a regular ring; 3） R is a reduced ring if and only if for each e∈E（R） and a∈N（R）, thereexists an element x of R such that ae=eaxae; 4） R is a strongly regular ring if and only if for each a,b∈R, there exists x∈R such that ab=baxab.