Left ideals in matrix near-rings

Meyer, J. H.

doi:10.1080/00927878908823791

Cited by 9 publications

(12 citation statements)

References 2 publications

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“…The weight of a matrix map A is basically the minimum number of elementary matrix maps needed to construct A. See [8] for a more detailed account of the notion of weight. First of all, for all r ∈ R and x, y ∈ R 2 , we have…”

Section: Near-rings Of Matrix Maps Over Integral Planar Near-ringsmentioning

confidence: 99%

“…Each of the A i and the B j should be seen as an expression consisting of elementary matrix maps and opening and closing parentheses in appropriate positions. In [8], it was shown exactly how to determine those f r ij in these expressions that act 'first' on the components of vectors in R 2 . For example, in A = f r1…”

Section: Introductionmentioning

confidence: 99%

“…The positions of these elementary maps in the expression A that act first are denoted by the set N A . See [8] for a detailed discussion about this.…”

Section: Introductionmentioning

confidence: 99%

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Matrix maps over planar near-rings

Wen¹,

Meyer²,

Wendt³

2010

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Following a method by Meldrum and van der Walt, near-rings of matrix maps are defined for general near-rings, not necessarily with identity. The influence of one-sided identities is discussed. When the base near-ring is integral and planar, the near-ring of matrix maps is shown to be simple. Various types of primitivity of the near-ring of matrix maps are discussed when the base near-ring is planar but not integral. Finally, an open problem concerning bijective matrix maps is solved.

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Section: Near-rings Of Matrix Maps Over Integral Planar Near-ringsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Matrix maps over planar near-rings

Wen¹,

Meyer²,

Wendt³

2010

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…For any k, lg/cgw, SC k is defined to be the left ideal of M n (R) generated by the matrix f\ k . We also define In Meyer [6] it is shown that if F is a near-field, then, with R replaced by F in the foregoing, J( k is a maximal left ideal of M) n (F). Moreover, it is shown that (l)).…”

Section: O-primitivitymentioning

confidence: 99%

“…Since then several papers ( [8,12,11,13,6,2,3]) and theses ( [7,1]) were devoted to matrix near-rings and as this field of study is still very immature, many more publications are expected to follow.…”

Section: Introductionmentioning

confidence: 99%

Left ideals and 0-primitivity in matrix near-rings

Meyer

1992

Proceedings of the Edinburgh Mathematical Society

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Maximal left ideals in matrix rings were studied by Stone [10]. Similar results are not necessarily valid in the general near-ring case and one of the objectives of this paper is to study these differences. Furthermore, although much is known about 2-primitivity in general matrix near-rings (Van der Walt [11]), quite the opposite is true for 0-primitivity and the other objective of this paper is to present some results on 0-primitivity in matrix near-rings in certain restricted cases.1980 Mathematics subject classification (1985 Revision): 16A76. IntroductionMatrix near-rings were introduced in 1984 by Meldrum and Van der Walt [5]. Since then several papers ([8, 12, 11, 13, 6, 2, 3]) and theses ([7,1]) were devoted to matrix near-rings and as this field of study is still very immature, many more publications are expected to follow.The purpose of this paper is to study 0-primitivity in matrix near-rings. A good survey on 2-primitivity in matrix near-rings over any zero-symmetric near-ring has been done by Van der Walt [11]. Some results on 0-primitivity are also contained in Abbasi, Meldrum and Meyer [2], but only for a very special class of near-rings, namely the weakly distributive d.g. near-rings. Because of some complexities, we could only manage to obtain certain results in restricted cases such as finite near-rings, or near-rings having the DCCR. It seems that a considerable amount of work still needs to be done to obtain similar results in the general zero-symmetric case.The first section merely introduces some of the basic definitions, results and techniques in matrix near-rings which will be used in this paper. For more details the interested reader should consult [5], [7] and [1]. Section 2 deals with maximal left ideals in matrix near-rings and the connections they have (or do not have) with maximal left ideals in the base near-ring. A counter-example is given to show that the near-ring case does not always necessarily follow the same pattern as in the ring case.The final section is devoted, for the greater part, to finite zero-symmetric near-rings and 0-primitivity. It becomes clear from this section that in order to have a reasonable understanding of modules over matrix near-rings, it is useful if one knows whether or not such modules can be embedded into a direct sum of finitely many copies of the additive group of the base near-ring. 173

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On the Development of Matrix Nearrings and Related Nearrings Over the Past Decade

Meyer

2001

Near-Rings and Near-Fields

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Left ideals in matrix near-rings

Cited by 9 publications

References 2 publications

Matrix maps over planar near-rings

Matrix maps over planar near-rings

Left ideals and 0-primitivity in matrix near-rings

On the Development of Matrix Nearrings and Related Nearrings Over the Past Decade

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