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Suppose $f,g$ are in $R[x_1,x_2,...,x_n]$ and $fg$ is homogeneous, then $f$ and $g$ are both homogeneous.

My argument is:

Considering $f$ and $g$ as monomials. suppose $fg$ is homogeneous and using the fact that deg$fg$ = deg$f$ + deg$g$, we can conclude that $f$ and $g$ are homogeneous.

Am I right?

Thanks

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    If $R$ is not an integral domain your problem is false. If $R$ is an integral domain, then the problem is okay, but your argument is wrong.2012-11-29

1 Answers 1