On Monochromatic Clean Condition on Certain Finite Rings

Kai An Sim (), Wan Muhammad Afif Wan Ruzali, Kok Bin Wong and Chee Kit Ho
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Kai An Sim: School of Mathematical Sciences, Sunway University, Petaling Jaya 47500, Malaysia
Wan Muhammad Afif Wan Ruzali: School of Mathematical Sciences, Sunway University, Petaling Jaya 47500, Malaysia
Kok Bin Wong: Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, Kuala Lumpur 50603, Malaysia
Chee Kit Ho: School of Mathematical Sciences, Sunway University, Petaling Jaya 47500, Malaysia

Mathematics, 2023, vol. 11, issue 5, 1-8

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 .

Keywords: finite commutative rings; monochromatic solution; monochromatic clean condition; generalised Ramsey theory (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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