On Monochromatic Clean Condition on Certain Finite Rings

Abstract

For a finite commutative ring R, let a,b,c∈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 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 Zn of integers modulo n.