Structure of Abelian rings

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.

Keywords

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

References

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

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

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

[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

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

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

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

[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

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

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

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

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

[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

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

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

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

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg