关于完全素环

ON FULLY PRIME RINGS

设Ｒ是群Ｇ分次环，若Ｒ的所有分次理想都是分次素理想，则称Ｒ是完全分次素环．证明了分次环Ｒ是完全分次素环当且仅当Ｒ的所有分次理想是幂等的，且所有分次理想作成的集合关于包含关系是全序的．设Ｇ是有限群，Ｒｅ是完全素环，Ｒ是强分次环当且仅当Ｒ＃Ｇ是完全素环．当Ｒ是强分次环时，若Ｒｅ是完全素环，则Ｒ是完全分次素环．

Let R be a graded ring by group G . R is said a fully graded prime ring, if every graded ideal of R is graded prime. A necessary and sufficient condition for a graded ring to be fully graded prime is that every graded ideal is idempotent and the set of graded ideals is totally ordered under inclusion. Let G be a finite group, R e is fully prime and R is strongly graded if and only if R # G * is fully prime. When R is strongly graded, R e is fully prime then R is fully graded prime. One example of such ring is constructed. It satisfies several conditions at the same time.