XLC环的若干性质

Some properties study on XLC rings

给出XLC环的定义,并研究XLC环的一些性质,主要证明了如下结果:1)R是交换约化环当且仅当G3(R)是XLC环;2)若R是XLC环,则R是直接有限环;3)设I_1,I_2是R的理想,R/I_1,R/I_2是XLC环且I_1∩I_2=0,则R/I_1∩I_2是XLC环;4)设R是XLC环,若a∈R是正则元,则a是强正则元.

In this paper,the authors introduce the notion of XLC rings and discuss some properties of XLC rings.We mainly show the following results:1)Ris a commutative reduced ring if and only if G3(R)is an XLC ring;2)If Ris an XLC ring,then Ris a directly finite ring;3)Let I_1,I_2 be ideals of Rsuch that R/I_1 and R/I_2 are XLC rings.Then R/I_1∩I_2 is an XLC ring;4)Let Rbe an XLC ring.If a∈Ris a regular element,then ais a strong regular element.