Clearing denominators relative to an independent module subset
= Clearing denominators relative to an independent module subset
{title2=$0\ne r\in R,\qquad rM\subseteq\langle T\rangle$}
Let a <finitely generated module> over an <integral domain> have a finite generating set $S$ and a maximal independent subset $T$. For each $s\notin T$, dependence of $T\cup\{s\}$ gives a nonzero coefficient $a_s$ with $a_ss\in\langle T\rangle$. The product of those coefficients is nonzero and carries the whole <module> into $\langle T\rangle$.