On monochromatic clean condition on certain finite rings
Document Type
Article
Publication Date
3-1-2023
Abstract
For a finite commutative ring R, let a, b, c is an element of R be fixed elements. Consider the equation ax + by = cz 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 chi of the elements of the ring R, there exist x, y, z is an element of R with chi(x) = chi(y) = chi(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 chi(i) = chi(j) for some i,j is an element of {x, y, z} where i &NOTEQUexpressionL; 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
Divisions
Science,MathematicalSciences
Publication Title
Mathematics
Volume
11
Issue
5
Publisher
MDPI
Publisher Location
ST ALBAN-ANLAGE 66, CH-4052 BASEL, SWITZERLAND