Near-Vector Spaces Constructed over Z&amp;lt;sub&amp;gt;&amp;lt;i&amp;gt;p&amp;lt;/i&amp;gt;&amp;lt;/sub&amp;gt;, for &amp;lt;i&amp;gt;p&amp;lt;/i&amp;gt; a Prime

Tefoetsile, Kelone; Kalunga, Jeromy; Zimba, Mathews; Muchinga, James; Chibeti, Saviour; Phiri, Henry M.

doi:10.4236/apm.2023.131002

APM

2023

DOI: 10.4236/apm.2023.131002

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Near-Vector Spaces Constructed over Zp, for p a Prime

Kelone Tefoetsile¹,

Jeromy Kalunga²,

Mathews Zimba³

et al.

Abstract: The purpose of this paper is to construct near-vector spaces using a result by Van der Walt, with p  for p a prime, as the underlying near-field. There are two notions of near-vector spaces, we focus on those studied by André [1]. These near-vector spaces have recently proven to be very useful in finite linear games. We will discuss the construction and properties, give examples of these near-vector spaces and give its application in finite linear games.

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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

Self Cite

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Near-Vector Spaces Constructed over Z<sub><i>p</i></sub>, for <i>p</i> a Prime

Cited by 1 publication

References 7 publications

An Introduction to the Theory of Matrix Near-Rings

An Introduction to the Theory of Matrix Near-Rings

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