MONOCHROMATIC ZERO DIVISOR SUMS IN CERTAIN FINITE COMMUTATIVE RINGS

Document Type : Original Manuscript

Authors

1 School of Mathematical Sciences, Sunway University, 47500 Bandar Sunway, Selangor, Malaysia.

2 Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, 50603 Kuala Lumpur, Wilayah Persekutuan Kuala Lumpur, Malaysia.

10.22044/jas.2025.14909.1877

Abstract

For a finite commutative ring R, let a, b, c ∈ R be fixed elements. Consider the equation aα+bβ = cγ where α, β and γ are zero divisors in R. We say that R satisfies the r-monochromatic zero divisor sum if for any r-colouring χ of the elements of the ring R, there exist α, β, γ ∈ R with χ(α) = χ(β) = χ(γ) such that the equation holds. We define ζ(a,b,c)(R) to be the least positive integer r such that R does not satisfy the r-monochromatic zero divisor sum. In this paper, we prove some results on ζ(a,b,c)(R) for various values of a, b, c. Special focus is given to the case where a = b = c = 1.

Keywords

References

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Journal of Algebraic Systems
Volume 14, Issue 4
September 2026
Pages 723-735

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