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FINITE UNITARY RINGS IN WHICH ALL SYLOW SUBGROUPS OF THE GROUP OF UNITS ARE CYCLIC

Part of: Conditions on elements Chain conditions, growth conditions, and other forms of finiteness

Published online by Cambridge University Press:  13 February 2019

M. AMIRI Open the ORCID record for M. AMIRI [Opens in a new window]  and
M. ARIANNEJAD Open the ORCID record for M. ARIANNEJAD [Opens in a new window]
M. AMIRI
Affiliation:
Departamento de Matemática-ICE-UFAM, 69080-900, Manaus-AM, Brazil email mohsen@ufam.edu.br
M. ARIANNEJAD*
Affiliation:
Department of Mathematics, University of Zanjan, Zanjan, Iran email arian@znu.ac.ir
*
✉arian@znu.ac.ir
Rights & Permissions [Opens in a new window]

Abstract

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We characterise finite unitary rings $R$ such that all Sylow subgroups of the group of units $R^{\ast }$ are cyclic. To be precise, we show that, up to isomorphism, $R$ is one of the three types of rings in $\{O,E,O\oplus E\}$, where $O\in \{GF(q),\mathbb{Z}_{p^{\unicode[STIX]{x1D6FC}}}\}$ is a ring of odd cardinality and $E$ is a ring of cardinality $2^{n}$ which is one of seven explicitly described types.

Keywords

finite ringgroup of unitsSylow subgroup

MSC classification

Primary: 16U60: Units, groups of units
Secondary: 16P10: Finite rings and finite-dimensional algebras

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 3 , June 2019 , pp. 403 - 412
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

Footnotes

This work has been partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) of the Ministry of Education of Brazil.

References

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