Structure of Abelian rings  

作　　者：Juncheol HAN Yang LEE Sangwon PARK 

机构地区：[1]Department of Mathematics Education, Pusan National University, Pusan 46241, Korea [2]Department of Mathematics, Dong-A University~ Pusan 49315, Korea

出　　处：《Frontiers of Mathematics in China》2017年第1期117-134,共18页中国高等学校学术文摘·数学（英文）

摘　　要：Let R be a ring with identity. We use J（R）, G（R）, and X（R） to denote the Jacobson radical, the group of all units, and the set of all nonzero nonunits in R, respectively. A ring is said to be Abelian if every idempotent is central. It is shown, for an Abelian ring R and an idempotent-lifting ideal N J（R） of R, that H has a complete set of primitive idempotents if and only if R/N has a complete set of primitive idempotents. The structure of an Abelian ring R is completely determined in relation with the local property when X（R） is a union of 2, 3, 4, and 5 orbits under the left regular action on X（R） by G（R）. For a semiperfect ring R which is not local, it is shown that if G（R） is a cyclic group with 2 ∈ G（R）, then R is finite. We lastly consider two sorts of conditions for G（R） to be an Abelian group.Let R be a ring with identity. We use J（R）, G（R）, and X（R） to denote the Jacobson radical, the group of all units, and the set of all nonzero nonunits in R, respectively. A ring is said to be Abelian if every idempotent is central. It is shown, for an Abelian ring R and an idempotent-lifting ideal N J（R） of R, that H has a complete set of primitive idempotents if and only if R/N has a complete set of primitive idempotents. The structure of an Abelian ring R is completely determined in relation with the local property when X（R） is a union of 2, 3, 4, and 5 orbits under the left regular action on X（R） by G（R）. For a semiperfect ring R which is not local, it is shown that if G（R） is a cyclic group with 2 ∈ G（R）, then R is finite. We lastly consider two sorts of conditions for G（R） to be an Abelian group.

关 键 词：Abelian ring regular group action local ring semiperfect ring finite ring Abelian group idempotent-lifting complete set of primitive idempotents 

分 类 号：O152.2[理学—数学] O152.7[理学—基础数学]