Structure of Abelian rings

作     者：Juncheol HAN Yang LEE Sangwon PARK 

作者机构：Department of Mathematics Education Pusan National University Pusan 46241 Korea Department of Mathematics Dong-A University~ Pusan 49315 Korea 

出 版 物：《Frontiers of Mathematics in China》 (中国高等学校学术文摘·数学（英文）)

年 卷 期：2017年第12卷第1期

页      面：117-134页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：supported by the Research Fund Program of Research Institute for Basic Sciences  Pusan National University  Korea  2015 

主　　题：Abelian ring regular group action local ring semiperfect ring finite ring Abelian group idempotent-lifting complete set of primitive idempotents 

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.