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

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 NJ(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.

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

1
Amitsur S A. A general theory of radicals III. Amer J Math, 1954, 76: 126–136

DOI

2
Antoine R. Nilpotent elements and Armendariz rings. J Algebra, 2008, 319: 3128–3140

DOI

3
Cohen J, Koh K. Half-transitive group actions in a compact ring. J Pure Appl Algebra, 1989, 60: 139–153

DOI

4
Goodearl K R. Von Neumann Regular Rings.London: Pitman, 1979

5
Goodearl K R, Warfield R B Jr. An Introduction to Noncommutative Noetherian Rings. Cambridge-New York-Port Chester-Melbourne-Sydney: Cambridge Univ Press, 1989

6
Grover K R, Khurana D, Singh S. Rings with multiplicative set of primitive idempotents. Comm Algebra, 2009, 37: 2583–2590

DOI

7
Han J, Lee Y, Park S. Semicentral idempotents in a ring. J Korean Math Soc, 2014, 51: 463–472

DOI

8
Han J, Park S. Additive set of idempotents in rings. Comm Algebra, 2012, 40: 3551–3557

DOI

9
Han J, Park S. Rings with a finite number of orbits under the regular action. J Korean Math Soc, 2014, 51: 655–663

DOI

10
Hirano Y, Huynh D V, Park J K. On rings whose prime radical contains all nilpotent elements of index two. Arch Math, 1996, 66: 360–365

DOI

11
Huh C, Kim H K, Lee Y. p.p. rings and generalized p.p. rings. J Pure Appl Algebra, 2002, 167: 37–52

DOI

12
Huh C, Lee Y, Smoktunowicz A. Armendariz rings and semicommutative rings. Comm Algebra, 2002, 30: 751–761

DOI

13
Hwang S U, Jeon Y C, Lee Y. Structure and topological conditions of NI rings. J Algebra, 2006, 302: 186–199

DOI

14
Jeon Y C, Kim H K, Lee Y, Yoon J S. On weak Armendariz rings. Bull Korean Math Soc, 2009, 46: 135–146

DOI

15
Jung D W, Kim N K, Lee Y, Yang S P. Nil-Armendariz rings and upper nilradicals. Internat J Algebra Comput, 2012, 22: 1–13 (1250059)

16
Kim N K, Lee Y. Armendariz rings and related rings. J Algebra, 2000, 223: 477–488

DOI

17
Lam T Y. A First Course in Noncommutative Rings.New York: Springer-Verlag, 1991

DOI

18
Lambek J. Lectures on Rings and Modules.London: Blaisdell Publ Co, 1966

19
Nicholson W K. Introduction to Abstract Algebra.Boston: PWS, 1998

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