I'm working on the following.
Let $R=R_0+R_1+ \cdots $ be a graded ring and $u$ a unit of $R_0$. Then the map $T_u$ defined by $T_u(x_0+x_1+ \cdots +x_n) = x_0+x_1u+ \cdots + x_n u^n$ is an automorphism of $R$ (this is clear). If $R_0$ contains an infinite field $k$, then an ideal $I$ of $R$ is homogenous iff $T_\alpha(I) = I$ for every $\alpha \in K^{\times}$.
I see that it fails for non-infinite fields, but I can't see what property to use of infinite fields to make this work. I have been thinking of maybe viewing I as a vector space over $k$, or using prime avoidance of some sort but it doesn't seem to do the trick. Any help would be most welcome.