Structure of Abelian rings

Autor: Juncheol Han, Yang Lee, Sangwon Park
Rok vydání: 2016
Předmět:
Zdroj: Frontiers of Mathematics in China. 12:117-134
ISSN: 1673-3576
1673-3452
DOI: 10.1007/s11464-016-0586-z
Popis: 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.
Databáze: OpenAIRE
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