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On Monochromatic Clean Condition on Certain Finite Rings

Author

Listed:
  • Kai An Sim

    (School of Mathematical Sciences, Sunway University, Petaling Jaya 47500, Malaysia
    These authors contributed equally to this work.)

  • Wan Muhammad Afif Wan Ruzali

    (School of Mathematical Sciences, Sunway University, Petaling Jaya 47500, Malaysia
    These authors contributed equally to this work.)

  • Kok Bin Wong

    (Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, Kuala Lumpur 50603, Malaysia
    These authors contributed equally to this work.)

  • Chee Kit Ho

    (School of Mathematical Sciences, Sunway University, Petaling Jaya 47500, Malaysia
    These authors contributed equally to this work.)

Abstract

For a finite commutative ring R , let a , b , c ∈ R be fixed elements. Consider the equation a x + b y = c z where x , y , and z are idempotents, units, and any element in the ring R , respectively. We say that R satisfies the r -monochromatic clean condition if, for any r -colouring χ of the elements of the ring R , there exist x , y , z ∈ R with χ ( x ) = χ ( y ) = χ ( z ) such that the equation holds. We define m ( a , b , c ) ( R ) to be the least positive integer r such that R does not satisfy the r -monochromatic clean condition. This means that there exists χ ( i ) = χ ( j ) for some i , j ∈ { x , y , z } where i ≠ j . In this paper, we prove some results on m ( a , b , c ) ( R ) and then formulate various conditions on the ring R for when m ( 1 , 1 , 1 ) ( R ) = 2 or 3, among other results concerning the ring Z n of integers modulo n .

Suggested Citation

  • Kai An Sim & Wan Muhammad Afif Wan Ruzali & Kok Bin Wong & Chee Kit Ho, 2023. "On Monochromatic Clean Condition on Certain Finite Rings," Mathematics, MDPI, vol. 11(5), pages 1-8, February.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:5:p:1107-:d:1077345
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    References listed on IDEAS

    as
    1. Kai An Sim & Kok Bin Wong, 2022. "Minimum Number of Colours to Avoid k -Term Monochromatic Arithmetic Progressions," Mathematics, MDPI, vol. 10(2), pages 1-10, January.
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