关于幺正则环

ON UNIT-REGULAR RINGS

设M是一R-模,End(M)为其自同态环.我们用直和分解的方式给出了End(M)是幺正则环的等价刻画,得到:End(M)是幺正则环当且仅当若有T=M1⊕N1=M2⊕N2,其中的M1∼=M2∼=M,则必有分解M1=M′1⊕(M1∩N2)和M2=M′2⊕M2′′,使得(M1∩N2)∼=M2′′且M′1⊕N2=M′2⊕N2.我们也对强正则环给出了类似的等价刻画.

Let M be an R-module and End(M)its endomorphism ring,it is proved that End(M)is a unit-regular ring if and only if whenever T=M1⊕N1=M2⊕N2 with M1∼=M2∼=M,there are decompositions M1=M′1⊕(M1∩N2)and M2=M′2⊕M2′′such that(M1∩N2)∼=M2′′and M′1⊕N2=M′2⊕N2.Strongly regular rings are described similarly.