We prove that whenever u,v∈N andA is a u× v matrix with integer entries and rank n,there is a u× n matrix B such that{Aec k:ec k∈Zv}∩ Nu={Bec x:ec x∈Nn}∩Nu.As a consequence we obtain the following result which answers a questionof Hindman, Leader, and Strauss: Let R be a subring of the rationals with 1∈ R and let S={x∈ R:x0}. If A is a finite matrix with rational entries,then there is a matrix B with no more columns than A such that the set of images ofB in S via vectors with entries from S is exactly the same as as the set ofimages of A in S via vectors with entries from R.We also show that the notion of image partition regularity is strictlystronger than that of weak image partition regularity in terms ofRamsey Theoretic consequences. That is, we show that for eachu≧ 3, there are no v and au× v matrix A such that for any ec y∈{Aec k:ec k∈Zv}∩Nu, the set of entriesof ec y form (in some order) a length u arithmetic progression.
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