群分次环的本原性及分次本原性

Primitivity and Graded Primitivity of Group Graded Rings

设A是一个用有限群G分次的环,本文给出了Smash积A＃G~*为本原环的一个判别准则,并证明了群分次环的每一个本原理想必定包含一个分次本原理想。作为一个推论,得到已被Cohen和Montgomery证实的Bergman猜想的另一个证明。此外还得到了A_1为本原环或单环的判别。

Let A be a group graded ring with a finite group G. First, we give a criterion for the smash product A#G to be primitive. Next, we show that every primitive ideal of a group graded ring must contain a graded primitive ideal. As a consequence, we give an other proof of the Bergman's conjucture answered by Cohen and Montgomery. Finally, we obtain criteria for A1 to be primitive or simple.