A concept of largeness of $CR$-sets

Let A be a u× v matrix with rational entries. In {key-6} N. Hindman and D. Strauss determined several characterizations of the property that whenever B is piecewise syndetic(J-set) in Z, \ xᵛ:Ax∈ Bᵘ\ is piecewise syndetic(J-set) in Zᵛ. In {key-1} V. Bergelson and D. Glasscock introduced combinatorially Rich sets(denoted $CR$-sets). Assume that for each a, there exists xᵛ such that Ax=āᵘ. In this article, we prove that \ yᵛ:Ay∈ Bᵘ\ is a $CR$-set in Zᵛ if B is $CR$-set in Z.