On invertible matrices over a near-field

Boykett, Tim; Wen, Kuei-Ann; Meyer, J. H.

doi:10.1016/j.jalgebra.2019.02.019

Journal of Algebra

2019

DOI: 10.1016/j.jalgebra.2019.02.019

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On invertible matrices over a near-field

Tim Boykett

Kuei-Ann Wen

J. H. Meyer

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2023

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An Introduction to the Theory of Matrix Near-Rings

Chibeti¹,

Kyapwanyama²

2023

APM

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Matrix rings are prominent in abstract algebra. In this paper we give an overview of the theory of matrix near-rings. A near-ring differs from a ring in that it does not need to be abelian and one of the distributive laws does not hold in general. We introduce two ways in which matrix near-rings can be defined and discuss the structure of each. One is as given by Beildeman and the other is as defined by Meldrum. Beildeman defined his matrix near-rings as normal arrays under the operation of matrix multiplication and addition. He showed that we have a matrix near-ring over a near-ring if, and only if, it is a ring. In this case it is not possible to obtain a matrix near-ring from a proper nearring. Later, in 1986, Meldrum and van der Walt defined matrix near-rings over a near-ring as mappings from the direct sum of n copies of the additive group of the near-ring to itself. In this case it can be shown that a proper near-ring is obtained. We prove several properties, introduce some special matrices and show that a matrix notation can be introduced to make calculations easier, provided that n is small.

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An Introduction to the Theory of Matrix Near-Rings

Chibeti¹,

Kyapwanyama²

2023

APM

View full text Add to dashboard Cite

show abstract

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On invertible matrices over a near-field

Cited by 1 publication

References 3 publications

An Introduction to the Theory of Matrix Near-Rings

An Introduction to the Theory of Matrix Near-Rings

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