{
  "id": "2405.07019",
  "version": "v1",
  "published": "2024-05-11T13:52:47.000Z",
  "updated": "2024-05-11T13:52:47.000Z",
  "title": "Classification of Integral Domains by combinatorially rich sets and extension of Goswami's theorem",
  "authors": [
    "Pintu Debnath",
    "Sourav Kanti Patra",
    "Shameek Paul"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In \\cite{Fi} A. Fish proved that if $E_{1}$ and $E_{2}$ are two subsets of $\\mathbb{Z}$ of positive upper Banach density, then there exists $k\\in\\mathbb{Z}\\setminus{0}$ such that $k\\cdot\\mathbb{Z}\\subset\\left(E_{1}-E_{1}\\right)\\cdot\\left(E_{2}-E_{2}\\right)$. In \\cite{G}, S. Goswami proved the same but a fundamental result on the set of prime numbers $\\mathbb{P}$ in $\\mathbb{N}$ and proved that for some $k\\in\\mathbb{N}$, $k\\cdot\\mathbb{N}\\subset\\left(\\mathbb{P}-\\mathbb{P}\\right)\\cdot\\left(\\mathbb{P}-\\mathbb{P}\\right)$. To do so, Goswami mainly proved that the product of an $IP^{\\star}$-set with an $IP_{r}^{\\star}$-set contains $k\\mathbb{N}$. This result is very important and surprising to mathematicians who are aware of combinatorially rich sets. In this article, we extend Goswami's result to large Lntegral Domain, that behave like $\\mathbb{N}$ in the sense of some combinatorics. we prove that for a combinatorially rich ($CR$-set) set, $A$, for some $k\\in\\mathbb{N}$, $k\\cdot\\mathbb{N}\\subset\\left(A-A\\right)\\cdot\\left(A-A\\right)$. we provide a new proof that if we partition a large Integral Domain, $R$, into finitely many cells, then at least one cell is both additive and multiplicative central, and we prove the converse part, which is a new and unknown result.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-11T13:52:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D10",
      "22A15",
      "54D35"
    ],
    "keywords": [
      "combinatorially rich sets",
      "goswamis theorem",
      "classification",
      "large integral domain",
      "large lntegral domain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}