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Suppose, $S$ is a ring, module-finite over a subring $R$. Let $s_1,...,s_n$ be a minimal set of generators of $S$ as an $R$-module. Can we have, $0=t_1s_1+...+t_ns_n$ for some $t_1,...,t_n\in R$, not all zero?

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In general (and typically) the answer is "yes".

Consider an integral domain $R$ with field of fractions $K$ and an element $x\in K$ which is integral over $R$. Then the ring $S:=R[x]\subseteq K$ is module-finite over $R$.

If $S\neq R$, that is if $x\not\in R$, then as an $R$-module $S$ is generated by at least two elements. Otherwise $S=Ry$ for some $y\in S$. Since $S$ is a ring one gets $y^2=ry$ for some $r\in R$. Consequently $y(y-r)=0$, hence $y\in R$ and thus $R=S$.

A minimal set of generators of $S$ as an $R$-module thus consists of two or more elements. If such a set is linearly independent over $R$, then it is linearly independent over $K$. However a $K$-linearly independent subset of $K$ consists of exactly one element. Consequently every minimal set of generators is linearly dependent over $R$.

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