### 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 language | English |
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Pages (from-to) | 226-235 |

Number of pages | 10 |

Journal | Journal of Algebra |

Volume | 232 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2000 Oct 1 |

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### 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.

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 232, no. 1, pp. 226-235. https://doi.org/10.1006/jabr.2000.8397

}

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 -