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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

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As YACP we need $R$ to be a domain.

The proof you give is insufficient because the "degree" ignores stuff of lower degree. How do you know $f$ and $g$ don't have stuff in lower degree that cancels out?

For example, if we worked over the ring $\mathbb{C}[t]/t^2$, then we have $ (x+t)(x-t) = x^2$

(Notice that your "proof" doesn't mind if $R$ is a domain or not, so we need to use that.)

Instead, it may be useful to look at the terms of lowest total degree in $f$ and $g$ and look what happens to their product in $fg$. (An ordering on monomials within a given total degree might make the argument slicker.)

If that's not enough of a hint to work it out I'll add more later.