中文摘要：

设R为一个非Abel的半Abelianπ-正则环,证明了下述条件等价：1）R仅有2个极大理想;2）Id（R）-{1}是本原的;3）E（R）={0,1}且对于e∈S0r（R）,f∈S0l（R）均有ef=0.进一步证明了如果S0l（R）R与RS0r（R）均为R的极大理想,那么R同构于一个正交准正则环与一个Abelianπ-正则环的亚直接和。

英文摘要：

Assume that theπ-regular ring Ris non-Abelian and semi-Abelian.The following conditions are equivalent：1）there exist just two maximal ideals in R.2）Id（R）-{1}is primitive.3）E（R）={0,1}and ef=0for any e∈S0r（R）,f∈S0l（R）.Furthermore,supposing that S0l（R）Rand RS0r（R）are both maximal ideals of R,we can prove that R will be isomorphic to the subdirect sum of an orthogonal pri-regular ring and an Abelianπ-regular ring.