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