Classification of Integral Domains by combinatorially rich sets and extension of Goswami's theorem

In Fi A. Fish proved that if E₁ and E₂ are two subsets of Z of positive upper Banach density, then there exists k0 such that k (E₁-E₁) (E₂-E₂). In G, S. Goswami proved the same but a fundamental result on the set of prime numbers P in N and proved that for some k, k (P-P) (P-P). To do so, Goswami mainly proved that the product of an IP^-set with an IPₑ^-set contains kN. 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 N in the sense of some combinatorics. we prove that for a combinatorially rich (CR-set) set, A, for some k, k (A-A) (A-A). 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.