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.

## 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 |

### Actions (login required)

View Item |