Sim, Kai An and Wan Ruzali, Wan Muhammad Afif and Wong, Kok Bin and Ho, Chee Kit (2023) On monochromatic clean condition on certain finite rings. Mathematics, 11 (5). ISSN 2227-7390, DOI https://doi.org/10.3390/math11051107.
Full text not available from this repository.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.
| Item Type: | Article |
|---|---|
| Funders: | UNSPECIFIED |
| Uncontrolled Keywords: | Finite commutative rings; Monochromatic solution; Monochromatic clean condition; Generalised Ramsey theory |
| Subjects: | Q Science > Q Science (General) Q Science > QA Mathematics |
| Divisions: | Faculty of Science Faculty of Science > Institute of Mathematical Sciences |
| Depositing User: | Ms Zaharah Ramly |
| Date Deposited: | 15 Jul 2024 08:06 |
| Last Modified: | 15 Jul 2024 08:06 |
| URI: | http://eprints.um.edu.my/id/eprint/38556 |
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