Comparison estimates for linear forms in additive number theory

Let $R$ be a commutative ring $R$ with $1_R$ and with group of units $R^{\times }$. Let $\mathrm{\Phi }=\mathrm{\Phi }(t_1,\dots ,t_h)={\sum }_{i=1}^h{\phi }_i{t}_i$ be an $h$-ary linear form with nonzero coefficients $\varphi_1,\ldots, \varphi_h \in R$\mathrm{.}Let$M$bean$R$-module\mathrm{.}Foreverysubset$A$of$M$,theimageof$A$under$\Phi$ is $$\mathrm{\Phi }(A)=\{\mathrm{\Phi }(a_1,\dots ,a_h):(a_1,\dots ,a_h)\in A^h\}\mathrm{.}$$ For every subset $I$ of $\{1,2,\dots ,h\}$, there is the subset sum $s_I={\sum }_{i\in I}{\phi }_i\mathrm{.}$ Let $ $\mathcal{S}$ (\Phi) = {s_I: \emptyset \neq I \subseteq {1,2,\ldots, h} }. $Theorem\mathrm{.}Let$\Upsilon(t_1,\ldots, t_g) = \sum_{i=1}^g \upsilon_it_i$and$\Phi(t_1,\ldots, t_h) = \sum_{i=1}^h \varphi_it_i$ be linear forms with nonzero coefficients in the ring $R$. If ${0, 1} \subseteq $\mathcal{S}$ (\Upsilon)$and$$\mathcal{S}$ (\Phi) \subseteq R^{\times}$,thenforevery$\varepsilon > 0$and$c > 1$thereexistafinite$R$-module$M$with$|M| > c$andasubset$A$of$M$suchthat$\Upsilon(A \cup {0}) = M$and$|\Phi(A)| < \varepsilon |M|$.