Let R be a commutative ring R with 1R and with group of units R×. Let Φ=Φ(t1,…,th)=∑i=1hφiti be an h-ary linear form with nonzero coefficients φ1,…,φh∈R. Let M be an R-module. For every subset A of M, the image of A under Φ isΦ(A)={Φ(a1,…,ah):(a1,…,ah)∈Ah}. For every subset I of {1,2,…,h}, there is the subset sumsI=∑i∈Iφi. Let S(Φ)={sI:∅≠I⊆{1,2,…,h}}.TheoremLetϒ(t1,…,tg)=∑i=1gυitiandΦ(t1,…,th)=∑i=1hφitibe linear forms with nonzero coefficients in the ring R. If{0,1}⊆S(ϒ)andS(Φ)⊆R×, then for everyε>0andc>1there exist a finite R-module M with|M|>cand a subset A of M such thatϒ(A∪{0})=Mand|Φ(A)|<ε|M|.