# <mtext>Content</mtext> ( f g ) </mrow> &#x2282;<!-- ⊂ -->

$\text{Content}\left(fg\right)\subset \text{Content}\left(f\right)\text{Content}\left(g\right)\subset \text{rad}\left(\text{Content}\left(fg\right)\right)$
to deduce that if $\text{Content}\left(f\right)$ contains a nonzerodivisor of $R$, then $f$ is nonzerodivisor of $S=R\left[{x}_{1},...,{x}_{r}\right]$.
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Elias Flores
By hypothesis, there is some nonzero $h\in R$ such that $fh=0$. By replacing $R$ with the subring of $R$ generated by the homogeneous parts of $f$ and $h$, we may assume that $R$ is finitely generated; in particular, $R$ is a Noetherian ring by the Hilbert basis theorem. Since $f$ is a zerodivisor, it is contained in some associated prime $P\in {\mathrm{Ass}}_{R}\left(R\right)$. By the above lemma, $P$ is the annihilator of a homogeneous element. The claim follows.