Structure of Abelian rings
Structure of Abelian rings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.