A concept of largeness of monochromatic sums and products in large ideal domain

An infinite integral domain $R$ is called a large ideal domain (LID) if every nontrivial ideal of $R$ has finite index in $R$. Recently, N. Hindman and D. Strauss have established a refinement of Moreira’s theorem for the set of natural numbers and infinite fields. In this article, we prove the same result of N. Hindman and D. Strauss for large ideal domains (LID) and a polynomial extension.