On Monochromatic Clean Condition on Certain Finite Rings

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