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

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

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
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Abstract

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
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. 

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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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