Frontiers of Mathematics in China >

2017 , Vol. 12 >Issue 1: 117 - 134

DOI: https://doi.org/10.1007/s11464-016-0586-z

RESEARCH ARTICLE

Structure of Abelian rings

  • Juncheol HAN 1 ,
  • Yang LEE , 1 ,
  • Sangwon PARK 2
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  • 1. Department of Mathematics Education, Pusan National University, Pusan 46241, Korea
  • 2. Department of Mathematics, Dong-A University, Pusan 49315, Korea

Received date: 09 Sep 2015

Accepted date: 03 Mar 2016

Published date: 01 Feb 2017

Copyright

2016 Higher Education Press and Springer-Verlag Berlin Heidelberg
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Abstract

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 R 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.

Key words： Abelian ring; regular group action; local ring; semiperfect ring; finite ring; Abelian group; idempotent-lifting; complete set of primitive idempotents

Cite this article

Juncheol HAN , Yang LEE , Sangwon PARK . Structure of Abelian rings[J]. Frontiers of Mathematics in China, 2017 , 12(1) : 117 -134 . DOI: 10.1007/s11464-016-0586-z

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