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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.

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    This is exercise $1$3.1 from Matsumura, *Commutative Ring Theory*.2013-01-31

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The usual way to prove this is to use Vandermonde determinants. To deal with all possible cases, one needs to have invertible Vandermonde determinants of all sizes, and for that one needs an infinite field.

By the way, this has nothing to do with the fact that $R$ is a ring or $I$ an ideal. It is in fact true that if $V=\bigoplus_{n\geq0}V_n$ is a graded vector space on which you have all those endomorphisms too, and $W\subseteq V$ is a subspace which is invariant, then $W$ is a graded subspace, that is, it is generated by the homogeneous elements it contains.

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    Indeed, that's it.2012-09-29