Some properties on rings with units satisfying a group identity

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

For convenience, a ring with units satisfying a group identity will be called a GI-ring. We show that GI-rings have the following properties which are also properites of PI-rings. (1) Any GI-ring is Dedekind finite (von Neumann finite). (2) Nilpotent elements of a semiprimitive GI-ring have bounded index. (3) The Kurosh problem has a positive answer for GI-algebras, namely, any algebraic GI-algebra is locally finite. We also study Hartley's problem for algebraic GI-algebras.

Original languageEnglish
Pages (from-to)226-235
Number of pages10
JournalJournal of Algebra
Volume232
Issue number1
DOIs
Publication statusPublished - 2000 Oct 1

Fingerprint

Ring
Unit
Algebra
Nilpotent Element

Keywords

  • Algebraic algebras
  • Dedekind finite
  • Group identities
  • Kurosh problem
  • Matrix units
  • Polynomial identities
  • Semiprimitive rings
  • semiprime rings

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Some properties on rings with units satisfying a group identity. / Liu, Chia-Hsin.

In: Journal of Algebra, Vol. 232, No. 1, 01.10.2000, p. 226-235.

Research output: Contribution to journalArticle

@article{a2f3718c8c274fd3aa6e0756e6577f86,
title = "Some properties on rings with units satisfying a group identity",
abstract = "For convenience, a ring with units satisfying a group identity will be called a GI-ring. We show that GI-rings have the following properties which are also properites of PI-rings. (1) Any GI-ring is Dedekind finite (von Neumann finite). (2) Nilpotent elements of a semiprimitive GI-ring have bounded index. (3) The Kurosh problem has a positive answer for GI-algebras, namely, any algebraic GI-algebra is locally finite. We also study Hartley's problem for algebraic GI-algebras.",
keywords = "Algebraic algebras, Dedekind finite, Group identities, Kurosh problem, Matrix units, Polynomial identities, Semiprimitive rings, semiprime rings",
author = "Chia-Hsin Liu",
year = "2000",
month = "10",
day = "1",
doi = "10.1006/jabr.2000.8397",
language = "English",
volume = "232",
pages = "226--235",
journal = "Journal of Algebra",
issn = "0021-8693",
publisher = "Academic Press Inc.",
number = "1",

}

TY - JOUR

T1 - Some properties on rings with units satisfying a group identity

AU - Liu, Chia-Hsin

PY - 2000/10/1

Y1 - 2000/10/1

N2 - For convenience, a ring with units satisfying a group identity will be called a GI-ring. We show that GI-rings have the following properties which are also properites of PI-rings. (1) Any GI-ring is Dedekind finite (von Neumann finite). (2) Nilpotent elements of a semiprimitive GI-ring have bounded index. (3) The Kurosh problem has a positive answer for GI-algebras, namely, any algebraic GI-algebra is locally finite. We also study Hartley's problem for algebraic GI-algebras.

AB - For convenience, a ring with units satisfying a group identity will be called a GI-ring. We show that GI-rings have the following properties which are also properites of PI-rings. (1) Any GI-ring is Dedekind finite (von Neumann finite). (2) Nilpotent elements of a semiprimitive GI-ring have bounded index. (3) The Kurosh problem has a positive answer for GI-algebras, namely, any algebraic GI-algebra is locally finite. We also study Hartley's problem for algebraic GI-algebras.

KW - Algebraic algebras

KW - Dedekind finite

KW - Group identities

KW - Kurosh problem

KW - Matrix units

KW - Polynomial identities

KW - Semiprimitive rings

KW - semiprime rings

UR - http://www.scopus.com/inward/record.url?scp=0034288183&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034288183&partnerID=8YFLogxK

U2 - 10.1006/jabr.2000.8397

DO - 10.1006/jabr.2000.8397

M3 - Article

AN - SCOPUS:0034288183

VL - 232

SP - 226

EP - 235

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 1

ER -