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- Title
On Monochromatic Clean Condition on Certain Finite Rings.
- Authors
Sim, Kai An; Wan Ruzali, Wan Muhammad Afif; Wong, Kok Bin; Ho, Chee Kit
- 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.
- Subjects
FINITE rings; RAMSEY numbers; IDEMPOTENTS; RAMSEY theory; INTEGERS
- Publication
Mathematics (2227-7390), 2023, Vol 11, Issue 5, p1107
- ISSN
2227-7390
- Publication type
Article
- DOI
10.3390/math11051107